•NRLF 


B    3    1ME 


REESE    LIBRARY 


UNIVERSITY   OF  CALIFORNIA. 

Received..  U/M> 

Accessions  No.<&&/^¥       Shelf  Nu.   . 


WORKS   ON 

DESCRIPTIVE  GEOMETRY, 

AND  ITS  APPLICATIONS  TO 

ENGINEERING,  MECHANICAL  AND  OTHER  INDUSTRIAL  DRAWING. 
By  S.  EDWARD  WARREN,  C.E. 


I.   ELEMENTARY  WORKS. 

1.  PRIMARY  GEOMETRY.     An    introduction   to  geometry  as 
usually  presented ;    and   designed,  first,  to   facilitate  an    earlier 
beginning   of  the   subject,  and,  second,  to  lead  to  its  graphical 
applications    in  manual  and   other  elementary  schools.       With 
numerous  practical  examples  and  cuts.    Large  12mo,  cloth,  80c. 

2.  FREE-HAND  GEOMETRICAL  DRAWING,  widely  and  variously 
useful  in  training  the  eye  and  hand  in  accurate  sketching  of  plane 
and  solid  figures,  lettering,   etc.     12  folding  plates,  many  cuts. 
Large  12mo,  cloth,  $1.00. 

3.  DRAFTING  INSTRUMENTS  AND  OPERATIONS.  A  full  descrip- 
tion of  drawing  instruments  and  materials,  with  applications  to 
useful  examples ;   tile  work,  wall  and  arch  faces,  ovals,  etc.     7 
folding  plates,  many  cuts.     Large  12mo,  cloth,  $1.25. 

4.  ELEMENTARY  PROJECTION  DRAWING.  Fully  explaining,  in 
six  divisions,  the  principles  and  practice  of  elementary  plan  and 
elevation  drawing  of  simple  solids  ;  constructive  details  ;~  shadows  ; 
isometrical  drawing ;   elements   of  machines ;   simple   structures. 
24  folding  plates,  numerous  cuts.    Large  12mo,  cloth,  $1.50. 

This  and  No.  3  are  especially  adapted  to  scientific,  preparatory, 
and  manual-training  industrial  schools  and  classes,  and  to  all 
mechanics  for  self-instruction. 

5.  ELEMENTARY    PERSPECTIVE.     With    numerous    practical 
examples,  and  every  step  fully  explained.     Numerous  cuts.    Large 
12mo,  cloth,  $1.00. 


6.  PLANE  PROBLEMS  on  the  Point,  Straight  Line,  and  Circle. 
225  problems.  Many  on  Tangencies,  and  other  useful  or  curious 
ones.  150  woodcuts,  and  plates.  Large  12mo,  cloth,  $1.25. 

II.  HIGHER  WORKS. 

1.  THE  ELEMENTS  OF  DESCRIPTIVE   GEOMETRY,    SHADOWS 
AND  PERSPECTIVE,  with  brief  treatment   of  Trehedrals ;  Trans- 
versals; and  Spherical,  Axonometric,  and  Oblique  Projections;  and 
many  examples  for  practice.    24  folding  plates.    8vo,  cloth,  $3.50. 

2.  PROBLEMS,  THEOREMS,  AND  EXAMPLES   IN   DESCRIPTIVE 
GEOMETRY.     Entirely  distinct  from  the  last,  with  115  problems, 
embracing   many  useful   constructions ;    52   theorems,   including 
examples  of  the  demonstration  of  geometrical  properties  by  the 
method  of  projections  ;  and  many  examples  for  practice.     24  fold- 
ing plates.     8vo,  cloth,  $2.50. 

3.  GENERAL   PROBLEMS   IN    SHADES   AND   SHADOWS,   with 
practical  examples,  and  including  every  variety  of  surface.     15 
folding  plates.    8vo,  cloth,  $3.00. 

4.  GENERAL  PROBLEMS   IN   THE   LINEAR  PERSPECTIVE  OF 
FORM,  SHADOW,   AND  EEFLECTION.   A  complete  treatise  on  the 
principles  and  practice  of  perspective  by  various  older  and  recent 
methods  ;  in  98  problems,  24  theorems,  and  with  17  large  plates. 
Detailed  contents,  and  numbered  and  titled  topics  in  the  larger 
problems,  facilitate  study  and  class  use.     Revised  edition,  correc- 
tions, changes  and  additions.    17  folding  plates.    8vo,  cloth,  $3.50. 

5.  ELEMENTS  OF  MACHINE  CONSTRUCTION   AND   DRAWING. 
73  practical  examples  drawn  to  scale  and  of  great  variety  ;  besides 
30  problems  and  31  theorems  relating  to  gearing,  belting,  valve- 
motions,  screw-propellers,  etc.     2  vols.,  8vo,   cloth,  one  of  text, 
one  of  34  folding  plates.    $7.50. 

6.  PROBLEMS  IN  STONE  CUTTING.     20  problems,  with  exam- 
ples for  practice  under  them,  arranged  according  to  dominant 
surface  (plane,  developable,  warped  or  double-curved)  in  each,  and 
embracing  every  variety  of  structure ;  gateways,    stairs,    arches, 
domes,  winding  passages,  etc.     Elegantly  printed  at  the  Riverside 
Press.     10  folding  plates.    8vo,  cloth,  $2.50. 


SCIENCE     X>  I?  '•  A.  W  I  INT  O- 


A  MANUAu 


OF 


ELEMENTAKY  PROBLEMS 


IN  THE 


LINEAR    PERSPECTIVE 


OP 


m  atifc  jlbatooto: 

C  ' 


OR  THE 

REPRESENTATION  OF  OBJECTS  AS  THEY  APPEAR, 

MADE   FROM   THE 

REPRESENTATION  OF  OBJECTS  AS  THEY  ARE 


In 

PART   I. PRIMITIVE    METHODS)    WITH    AN    INTRODUCTION. 

PART   II. — DERIVATIVE    METHODS)    WITH    SOME   NOTES   ON   AERIAL    PERSPECTIVE. 


BY  S.  EDWARD  WARREN,  C.  K, 

PBOFES8OR  OF  DESCRIPTIVE  OEOMKTRY,   ETC.,   IN  THE  REN8SELAEE  POLYTECHNIC  INSTITUTE 

AND   AUTUOE  OF  TUB  "  DRAFTSMAN'S  MANUAL  ;"  AND  "  GENERAL  PROBLEMS  O» 

DESCRIPTIVE  GEOMETRY." 


NEW  YORK  : 
JOHN    WILEY    &    SONS, 

15  ASTOR  PLACE. 
1888. 


Entered  according  to  Act  of  Congress,  fc;  the  year  eighteen  hundred  and  sixty-three,  by 
8.    EDWAED    WAKBEN, 

In  the  CWK  s  Office  of  the  District  Court  c  f  the  United  States  for  the  No-thern  District  -Jl 

New  York. 


CONTENTS. 


PAG  I 

PREFACE  . .  vi 


INTRODUCTION. 

CHAPTER  I. — Instruments  and  Materials 9 

Paper 9 

Support  of  Paper 9 

Pencils ? 9 

Rulers '. 9 

Compasses %  10 

Use  of  Compasses 10 

Irregular  Curves 10 

Indian  Ink 10 

CHAPTER  II. — Preliminary  Principles  and  Explanations 12 


PART  I. 

PRIMITIVE  METHODS. 

CHAPTER  I. — Definitions  and  General  Principles I 

"        II. — The  Elements  of  Projections 1 

"      III. — The  Construction  of  the  Perspectives  of  Objects  from  their  Pro- 
jections   24 

"      IV. — Real  Projections,  and  Perspectives  made  from  them 28 

Perspectives  of  Geometrical  Solids,  Art.  (68.) 32 

Example  1. — To  Find  the  Perspective  of  a  Vertical  Square 

Prism °2 

"         2. — To    Find    the    Perspective  of  a  Triangular 

Pyramid 85 


IV  CONTENTS. 

PA  en 
CHAPTER  Y  — Removal  of  Practical  Difficulties,  arising  from  the  confusion  of 

Projections  and  Perspectives 36 

§  I.  —First  Method. — Translation  forward  of  the  Perspective  Plane..  36 

Example  3. — To  Find  the  Perspective  of  a  Cube,  etc 38 

§  II.— Second  Method.— Use  of  Three  Planes 38 

Example  4.— To  Find  the  Perspective  of  an  Obelisk,  etc. . .  41 

VI. — Projections  and  Perspectives  of  Circles,  and  of  Bodies  having 

partly  or  wholly  curved  boundaries 4 

Example  5.— To  Find  the  Perspective  of  a  Circle,  lying  in 

the  horizontal  plane 43 

Of  Planes,  Arts.  (77-84.) 45-46 

"          6. — To  Find  the  Perspective  of  a  Cylinder,  stand- 
ing on  the  horizontal  plane 47 

44          7. — To  Find  the  Perspective  of  a  Cone,  standing 

on  the  horizontal  plane 50 

u          8. — Do.  of  a  Cone  whose  axis  is  parallel  to  the 

ground  line 52 

9. — Do.  of  a  Cone  whose  axis  is  parallel  to  the 

vertical  plane  only 64 

;<         10. — Do.  of  a  Cone  whose  axis  is  oblique  to  both 

planes  of  projection 55 

"        11.— To  Find  the  Perspective  of  a  Sphere 58 

First  Method  of  finding  the  apparent  contour.     58 
Second  Method        "  "  "  61 

"        12.— To  Find  the  Perspective  of  a  Concave  Cupola 

Roof. 63 

CHAPTER  VII. — Perspectives  of  Shadows 65 

General  Principles  and  Illustrations,  Arts.  (89-99.). 65 

Example  13. — To  Find  the  Perspective  of  the  Shadow  of 

a  Square  Abacus  on  a  Square  Pillar. 67 

«        14.— Do.  of  a  Triangular  Pyramid  upon  the  Hori- 

zontal  Plane 69 

tt        15. — Do.  of  a  Dormer  Window  upon  a  Roof. 71 


PART  II. 

DERIVATIVE  METHODS. 

Oe AFTER  I. — General  Principles  and  Illustrations 75 

Example  1.— To  Find  the  Vanishing  Point  of  Telegraph 

Wires,  etc 78 


CONTENTS.  V 

PAG1 

Example  2. — To  Find  the  Vanishing  Point  of  a  Perpendi- 
cular and  of  a  Diagonal 79 

Particular  Derivative  Methods,Arts.  (1 18-121.)  81 
«          3. — To  Find  the  Perspective  of  a  Straight  Line,  in 
any  position  oblique  to  both  planes  of  pro- 
jection, etc 82 

14          4-. — Do.  of  a  Tower  and  Spire 84 

Practical  Remarks,  Art.  (122.) 86 

«          5. — To  Find  the  Perspective  of  a  Cross  and  Pedestal  89 

CHAPTER  II. — Perspectives  of  Shadows 92 

Example  6. — To  Find  the  Vanishing  Poiat  of  Rays,  and  of 

their  Horizontal  Projections 92 

«          7. — TO  Find  the  Perspective  of  the  Shadow  of 

any  Vertical  Line  upon  the  Horizontal  Plane.  93 

CHAPTER  III. — Miscellaneous  Problems 96 

Example  8. — To  Find  the  Perspective  of  a  Pavement  of 
Squares,  whose  sides  are  parallel  to  the 

ground  line 96 

*«          9. — Do.  of  a  Pavement  of  Hexagons,  whose  sides 
make  angles  of  30°  and  90°  with  the  ground 

line 98 

««        10. — Do.  of  an  Interior 98 

«        11. — Do.  of  the  Shadows  in  an  Interior 102 

«        12.— Do.  of  a  Cabin 104 

"        13.— Do.  of  the  Shadow  of  a  Chimney  on  a  Roof. .  108 

CHAPTER  IV. — Pictures,  and  Aerial  Perspective 110 

Landscape  Outlines 110 

Landscape  Details. — Trees Ill 

Hills Ill 

Valleys Ill 

Ascent  and  Descent 112 

Level  of  the  Eye 112 

Reflections  in  "Water. 113 

Location  of  the  Centre  of  the  Picture.  113 

Do.  of  the  Perspective  Plane 114 

Shadows  of  Trees,  and  other  Vertical 

Objects 114 

Time  of  a  Given  Aspect 114 

Light  and  Shade 115 

Edges 116 

Color..                                                  .  11.6 


PREFACE. 


FOR  several  years  past,  while  teaching  a  comparatively  advanced 
course  on  Perspective,  embracing  some  of  its  higher  problems, 
I  have  cherished  a  purpose  to  compose  an  elementary  perspec- 
tive, for  genera]  use ;  which  should  be  clearly  demonstrative  at 
every  step,  and  also,  if  possible,  interesting  to  its  readers  ;  which 
should,  in  fact,  be  truly  popular,  without  being  empirical ;  and, 
on  the  other  hand,  perfectly  demonstrative,  without  being  too 
elevated.  In  other  words,  I  have  sought  to  make  my  work 
elementary,  not  in  the  sense  of  merely  stating  perspective  facts 
without  adequate  explanation,  but  in  the  sense  of  selecting  sim- 
ple, yet  widely  and  always  useful  examples,  and  then  fully 
explaining,  in  easy  order,  the  few  plain  principles  necessary  to 
the  solution  of  such  examples. 

An  exact  knowledge  of  perspective  is  indispensable  to  those 
who  would  make  exact  representations,  for  industrial  purposes, 
of  architectural  or  mechanical  structures  as  they  appear.  It  is 
highly  useful  to  those,  even,  who  practise  perspective  as  an 
ornamental  art,  in  the  making  of  pictures;  inasmuch  as  it 
nables  them  to  know  scientifically,  as  well  as  feel  sensibly, 
whether  their  drawings  are  correct  or  incorrect.  It  is  also 
interesting  to  the  amateur  judges  and  admirers  of  pictures,  as 
well  as  to  their  makers ;  and,  finally,  it  is  useless  to  none  who  are 
in  any  manner  engaged  with  the  arts  of  graphical  representation 
or  design. 

It  is  a  part  of  the  price  of  truth,  whereby  we  discover  its 
worth,  that  we  must  discover  the  truth  concerning  propriety  oj 


Vlll  PREFACE. 

arrangement  in  any  subject,  as  an  indispensable  condition  for 
its  successful  treatment.  A  course  of  research,  which  in  any 
degree  ignores  certain  elements  on  which  it  is  based,  cannot  but 
become  proportionately  involved  in  bewildering,  and,  both  to 
the  student  and  critic,  comfortless  confusion  and  intricacy.  It 
is  equally  true,  that  natural  progress  from  a  proper  starting 
point  cannot  fail  to  be  effectual  and  agreeable. 

As  the  truth  about  primary  industrial  utilities  for  daily  work- 
ing life,  is  more  elementary  and  fundamental  than  truth  about 
beauty  and  other  higher  utilities  for  the  adornment  of  higher 
life,  it  results,  in  the  present  instance,  that  "  Projections,"  which, 
usually  for  industrial  purposes,  represent  objects  as  they  are, 
in  form  and  size,  naturally  precede,  in  a  course  of  exact  study, 
"Perspective,"  which,  usually  for  pictorial  effect,  represents 
objects  as  they  appear. 

Perspectives,  or  drawings  of  objects  as  they  appear,  are  made, 
then,  from  Projections,  or  drawings  of  objects  as  they  are  ;  and 
which,  therefore,  are  competent  representatives  of  those  objects. 

The  study  of  projections  thus  properly  preceding  that  of 
perspective,  as  its  natural  foundation,  disadvantages  will 
unavoidably  arise  from  attempts  to  treat  of  exact  perspective, 
without  a  formal  preliminary  treatment  of  ppojections.  Hence, 
this  work,  while  complete  in  itself,  is  the  natural  successor,  for 
those  who  use  both,  of  my  "  Manual  of  Elementary  Geometrical 
Drawing  of  Three  Dimensions,"  in  which  objects  are  shown  in 
projection  only. 

It  is  unfortunate  for  learners,  that  a  subject  so  simple,  useful, 
and  attractive,  as  Perspective  is,  when  properly  treated,  should 
come  to  be  regarded  with  aversion,  merely  owing  to  defects  in 
its  treatment ;  the  chief  of  which  defects  is,  perhaps,  the  failure 
fully  to  exhibit  its  foundation  in  "projections." 

The  present  volume  is  an  attempt  to  expressly  present  Per- 
spective, as  founded  on  "Projections,"  and  in  this  respect  it 
differs,  more  or  less  noticeably,  from  numerous  elementary 
works  on  the  subject.  I  accordingly  hope  for  such  results,  in 


PREFACE. 


respect  to  ready  and  interested  understanding  of  the  subject, 
as  the  improved  treatment  of  it,  which  I  have  endeavored  to 
give,  leads  me  to  anticipate. 

The  construction  of  the  perspective  of  a  shadow  is,  from  first 
to  last,  a  problem  of  more  tediousness  and  complexity,  espe 
cially  as  applied  to  complex  objects  and  positions,  than  falls 
within  the  scope  of  an  elementary  work  like  the  present. 
Hence,  only  a  few,  and  quite  simple,  problems  in  perspectives  of 
shadows  have  been  inserted. 

The  simplest  conception,  and  resulting  definition,  of  the  per- 
spective of  a  point  is,  that  it  is  where  the  "visual  ray" 
through  the  point  pierces  the  plane  of  the  picture,  i.e.  the 
"  perspective  plane." 

The  method  of  construction  just  indicated,  and  here  adopted 
for  PART  I.,  does  away  with  the  whole  machinery  of  "  vanish- 
ing points,"  "perpendiculars,"  "diagonals,"  etc.;  and,  accord- 
ingly, these,  with  their  advantages,  are  briefly  explained  and 
illustrated,  in  PART  II.,  as  incidental  matters,  giving  rise 
to  derivative  methods  of  construction,  and  tending  to  aid  the 
reader  in  understanding  the  methods  usually  employed  by 
writers  on  perspective. 

In  a  proposed  future  general  work  on  perspective,  I  hope  to 
exhibit  more  fully  a  systematic  arrangement  of  all  its  methods, 
and  interesting  peculiarities  and  details. 


LINEAR  PERSPECTIVE. 


INTRODUCTION. 
CHAPTER  I. 

INSTRUMENTS   AND   MATERIALS. 

1.  PAPER. — For  elementary  practice,  thick  unruled  writing,  or 
tough  printing  paper,  will  answer.     For  nicer  work,  German  car- 
toon,  or  English  smooth  drawing  paper,  will  be  convenient ;  and  for 
exact  constructions,  in  lines  or  tints  of  Indian  Ink,  Whatman's 
drawing  paper  will  be  best. 

2.  Support  of  the  Paper. — For  slight  pencil  or  ink  sketches,  the 
paper  may  lie  flat  on  an  atlas,  or  a  few  thicknesses  of  smooth 
paper,  or  any  firm,  but  not  rigid  surface.     For  larger  and  exact  ink 
drawings,  the  paper  must  be  well  wet,  and  then  fastened  round 
the  edges  with  gum-arabic,  to  a  smooth  board.     Then,  when  dry, 
it  will  be  found  to  be  tightly  stretched. 

Great  care  should  be  taken  to  keep  paper  flat  and  smooth,  when 
not  stretched  as  just  described. 

3.  PENCILS. — For  sketching,  use  a  hard  pencil,  as  No.  4,  or  5,  of 
Faber's,  and  make  only  the  faintest  lines.     For  finishing  up  pencil 
drawings,  use  softer  and  blacker  pencils,  as  No.  3  for  well  defined 
objects,  and  Nos.  1  and  2  for  shadows,  foliage,  &c. 

In  pencilling  a  drawing  which  is  to  be  inked,  use  a  pencil 
sharpened  on  a  fine  file,  to  a  thin  edge,  rather  than  a  round  point, 
since  it  will  thus  keep  sharp  much  longer. 

4.  RULERS. — For  drawing  on  stretched  paper,  use  a  T  rule,  foi 
drawing  all  lines  from  right  to  left ;  and  a  right  angled  triangle,  for 
drawing  lines  perpendicular  to  these.     With  loose  paper,  use  a 
common  ruler  and  right  angled  triangle. 


10  LINEAR    PERSPECTIVE. 

5.  To  draw  parallels  in  any  oblique  position,  by  a  ruler  and  tri 
angle.    To  draw  through  p,  for  example,  a  parallel  to  ab.    Place 


FIG.  1. 

a  side  of  the  triangle  on  ab,  and  bring  up  the  ruler  ac,  as  shown 
in  Fig.  1.  Hold  the  ruler  fast,  and  slide  down  the  triangle  to  the 
position  (2)  when  pd  will  be  parallel  to  ab. 

6.  To  draw  perpendiculars  in  oblique  positions. — Slide  the  tri- 
angle, as  before,  till  de  passes  through  p,  then  d  being  a  right 
angle,  a  line  can  be  drawn  through  p,  and  perpendicular  to  ab. 

7.  COMPASSES. — For  drawing    ink   or   pencil  circles,  the    com- 
passes  should  have  movable  legs,  which  may  be  replaced  by  a 
drawing  pen,  or  pencil-holder. 

8.  In  using  the  compasses,  hold  them  by  the  joint,  with  the 
thumb  and  forefinger.     Then,   in  setting  off  distances  on  a  line, 
turn  them,  alternately,  on  one  side  and  the  other  of  the  line,  never 
taking  both  points  at  once  from  the  paper,  till  the  operation  is 
finished.     This  method  is  most  expeditious  and   accurate.     Like 
wise,  in  describing  a  circle,  the  whole  can  be  accomplished  with 
quick  and  uninterrupted  motion. 

9.  Irregular  Curves. — For  drawing  other  curves  than  circles, 
points  of  which  have  been  previously  constructed,  use  the  thin 
plate  of  wood  with   variously  curved  edges   and   openings,  and 
called  an  irregular  curve. 

10.  INDIAN  INK. — This   color,  when  of  good   quality,   is   of  a 
brownish  black,  and  is  prepared  in  polished  or  gilded  cakes,  fine- 
grained, and  usually  scented  with  musk  or  camphor.    It  is  prepared 
for  use,  like  other  water  colors,  by  touching  the  end  to  water,  an<q 


INSTRUMENTS    AND    MATERIALS.  1] 

rubbing   on   an  earthen  plate  or  tile.      When   enough  has    been 
ground  off,  wipe  the  cake  dry  to  prevent  its  crumbling. 

11.  This  ink,  when  thick,  may  be  applied  in  a  drawing-pen,  or 
brush,  so  as  to  make  black  lines,  or  surfaces.  When  diluted  with 
a  quantity  of  water,  tints  of  any  degree  of  lightness  may  be  quickly 
laid  on  the  paper  (stretched)  by  a  rapid  use  of  a  goose-quill  sized, 
or  larger  camel's  hair  brush. 


12  LINEAR    PERSPECTIVE. 


CHAPTER  II. 

PTHT.TJMINARY   PRINCIPLES    AND    EXPLANATIONS. 

12.  Sitting  by  a  window,  you  may  fix  your  attention  on  all 
that  you  see  through  one  of  its  panes — buildings  and  parts  thereof, 
trees,  roads,  fields,  woods,  streams  and  clouds. 

As  soon,  however,  as  you  give  attention,  both  to  the  pane  and 
to  what  you  see  through  it,  you  will  find  that,  by  looking  with  each 
eye  separately,  you  will  see  partly  different  sights  through  the 
same  pane.  Hence,  to  see  definitely  both  'the  pane  and  what  you 
see  through  it,  you  must  close  one  eye. 

13.  This  being  done,  you  might  paint  upon  the  glass  everything 
that  you  see  through  it,  just  where  you  see  it,  and  of  the  same 
shade  and  color.     A  perfect  picture,  in  every  respect,  of  all  seen 
through  the  glass,  from  one  point  of  sight,  might  thus  be  made  on 
the  pane.     Such  a  picture  would  be  called  the  perspective  of  the 
view  seen  through  the  pane. 

14.  I.  Hence  &  perspective  is  a  picture  which  shows  one  or  more 
objects  just  as  they  appear,  in  respect  both  to  form  and  color,  and 
as  seen  from  one  fixed  point  of  sight. 

15.  If  seated  quite  near  the  window,  you  will  observe  that  you 
cannot  see  all  that  is 'to  be  seen  through  it,  without  turning  the 
head ;  while  each  new  direction  of  sight  gives  you  at  least  a  partly 
new  view.     Also  each  new  position  of  the  eye  gives,  evidently,  a 
different  view  through  the  same  pane, 

16.  II.  Hence  any  single  perspective  drawing  should  embrace 
no  more  than  one  view,  that  is,  no  more  than  can  really  be  seen 
vhen  looking  in  one  direction  from  one  fixed  point  of  sight. 

17.  The  chief  exceptions  to  this   rule   are  in  panoramic  and 
architectural    interior   scene   painting,    which,  being   intended   to 
please  large  assemblies,  are  painted  from  several  points  of  sight,  or 
from  one  quite  remote  one. 

18.  All  this  being  understood,  suppose  you  are  in  a  field,  and 
dewing  a  distant  tree  through  a  framed  pane  of  glass,  held  at  a 

.axed  distance  from  the  eye.  As  yon  approach  the  tree  it  appears 
to  occupy  a  larger  and  larger  portion  of  the  glass ;  while,  as  you 
recede  from  it,  a  contrary  effect  is  produced. 


PRELIMINARY    PRINCIPLES    AND    EXPLANATIONS.  13 

III.  Hence  the  size  of  the  object,  in  a  picture,  depends  on  its 
distance  from  the  eye. 

19.  Again;  if  your  distance  from  the  tree  is  fixed,  the  nearer 
the  pane  is  carried  to  the  tree  the  more  completely  will  the  view 
of  the  tree  fill  it.     That  is— 

IY.  The  comparative  size  of  an  object  in  the  picture,  and  th° 
whole  picture,  depends  also  on  the  distance  of  the  picture  plan 
from  the  object. 

20.  Further,  if  two  trees,  at  equal  distances,  and  of  different 
sizes,  be  viewed  at  once  through  the  same  pane,  and  from  the  same 
fixed  position,  the  larger  one  will  cover  a  larger  space  on  the  pane, 
as  seen  through  it. 

Y.  Hence,  other  things  being  the  same,  the  size  of  an  object, 
in  the  picture,  depends  on  its  actual  size. 

21.  Once  more,  by  moving,  together  with  the  pane,  from  side 
to  side,  or  up  and  down,  the  tree  will  be  seen  through  different 
portions  of  the  pane,  when  seen  from   the  different  positions  so 
taken. 

YI.  That  is,  the  place  of  an  object  in  a  given  picture,  its  size 
and  distance  being  also  given,  depends  on  its  direction  from  the 
observer. 

22.  VII.  From  the  last  four  particulars,  we  now  conclude  that, 
in  order  to  represent  a  given  object  truly,  its  dimensions,  distance 
from  the  picture,  distance  from  the  eye,  and  direction  must  all  be 
known. 

In  other  words,  the  relative  position  of  the  eye,  the  picture,  and 
the  object,  and  the  size  of  the  latter,  must  be  known. 

23.  Returning  now  to  the  picture  painted  on  the  window  pane, 
each  point  of  that  picture  is  in  a  straight  line,  from  the  point  repre- 
sented, to  the  eye.     Such  a  line  is  called  a  visual  ray. 

VIII.  Hence  the  perspective  of  any  point  is  where  the  visual 
ray  from  that  point  meets  the  surface  of  the  picture. 

Finally,  the  following  general  principles  may  serve  to  connect; 
his  introductory  sketch,  which  embraces  the  primary  facts  of  per 
spective,  given  by  the  testimony  of  the  senses,  with  the  more  exact 
treatment  of  the  subject,  which  succeeds,  and  in  which  the  prin- 
ciples of  perspective,  based  upon  these  facts,  are  demonstrated. 

24.  Science  is  a  complete  body  of  truth,  whose  parts  are  naturally 

related  to  each  other ;  and  hence  may  be  expressed  by  a  systematic 

and  connected  statement  of  successive  particulars,  proceeding  in 

natural  order  from  primary  elements  to  complete  results. 

Perspective  science  is  such  a  body  of  truth,  relating  to  the 


14  LINEAR   PERSPECTIVE. 

manner  of  representing  objects  as  they  appear.  This  science  IP 
founded  on  the  simple  facts  of  vision  already  described,  and  which 
are  learned  by  observing  what  and  how  we  see. 

As  in  making  a  picture  itself,  its  outlines  and  most  conspicuous 
objects,  alone,  may  be  represented,  or  all  its  peculiarities  and 
details  may  also  be  included;  so  a  science  may  be  presented  in  its 
outlines  only,  or  in  entire  completeness. 

This  work  aims  to  exhibit  little  more  than  the  outlines  of  the 
subject  of  perspective,  but  yet  fully  enough  to  assist  any  one,  who 
desires  to  draw  ordinary  objects  as  a  business  or  pleasure,  to  do  so 
intelligently  and  accurately. 

We  now  proceed  to  unfold  the  elements  of  Perspective  from  the 
preceding  simple  facts  of  vision,  and  to  apply  those  elements  to 
practical  exercises  in  perspective  drawing. 


DEFINITIONS    AND    GENERAL   PRINCIPLES.  Iff 

PART  I 

PRIMITIVE     METHODS. 


CHAPTER  I. 

DEFINITIONS  AND  GENERAL  PRINCIPLES. 

25.  The  complete  perspective  of  an  object,  is  a  picture  of  it, 
which,  when  viewed  from  a  certain  point,  produces  the  same  image 
upon  the  eye  that  the   object  itself  does,  when  viewed  from  the 
game  point. 

Each  point  and  line  of  such  a  picture,  must,  when  suitably  placed 
between  the  eye  and  the  object,  exactly  cover  and  conceal  from 
view  the  corresponding  points  and  lines  of  that  object.  It  must 
also,  as  truly  as  art  will  allow,  present,  at  each  point,  the  same 
shade  and  color  that  is  exhibited  by  the  same  object. 

26.  Hence  perspective  embraces  two  branches:  the  perspective 
of  form,  called  linear  perspective  /  and  the  perspective  of  color  and 
gradations  of  shade,  called  aerial  perspective. 

27.  Aerial  perspective  is  an  imitative  art,  founded  on  extensive 
observation  of  nature,  and  on  the  science  of  optics. 

Linear  perspective  is  either  an  imitative  art,  or  an  art  of  exact 
geometrical  construction,  according  as  the  outlines  of  pictures  of 
given  objects  are  traced  by  the  eye,  or  constructed  with  instru- 
ments, according  to  geometrical  principles. 

28.  In  point  of  fact,  linear  perspective  is  practised  as  a  construc- 
tive art,  chiefly  in  its  application  to  regular  objects.     It  is  practised 
as  an  imitative  art,  mainly  in  the  drawing  of  irregular,  or  pic 
turesque  objects,  such    as  trees,  animals,  hills,   streams,  and    old 
buildings. 

We  will  next  inquire  into  the  natural  principles,  which  lead  to 
the  exact  construction  of  the  perspectives  of  objects. 

29.  The  eyes  are  so  related,  that  in  attempting  to  see,  with  both 
of  them  together,  objects  at  different  instances,  distinctly  and  at 
once,  we  see  these  objects  partly  double  (12). 

Hence  in  making  an  exact  picture  of  any  object,  we  suppose  it 


1C  LINEAR    PERSPECTIVE. 

to  be  viewed  with  one  eye,  or  that  the  two  eyes  are  reduced  to  a 
single  seeing  point,  called  the  point  of  sight  (14). 

30.  Objects  become  visible  by  means  of  rays  of  light,  reflected 
from  them  to  the  eye,  and  called  visual  rays  (23). 

31.  Kays  of  light  proceed  in  straight  lines;  as  is  proved  by  the 
fact  that  we  can  see  nothing  through  an  opaque  bent  tube. 

32.  The  visible  boundary  of  an  object  is  called  its  apparent  con- 
tour.    The  perspective  of  this  contour,  is  the  linear  perspective  of 
the  object. 

33.  Any  body,  having  a  vertex,  and  plane  sides,  is,  in  a  genera1 
sense,  called  a  Pyramid. 

Any  curved  surface,  having  a  vertex,  and  therefore  containing 
straight  lines  drawn  through  that  vertex,  is  a  Cone,  in  the  general 
sense  of  the  term. 

Hence  visual  rays,  from  all  points  of  the  apparent  contour  of  an 
object  to  the  eye,  form  a  pyramid,  or  a  cone — whose  vertex  is  the 
point  of  sight — according  as  the  object  is  bounded  by  straight  or 
by  curved  lines.  This  being  understood,  this  pyramid  and  cone 
are,  for  the  sake  of  brevity,  called  indifferently  the  visual  cone. 

34.  Next,  conceive  a  plane  to  intersect  the  visual  cone,  anywhere 
between  its  vertex,  that  is  the  eye,  and  its  base,  that  is  the  object. 
This  plane  will  cut  from  the  cone  a  figure  which  will  exactly  conceal 
from  the  eye  at  its  vertex,  the  apparent  contour  of  the  original 
object.     That  is  (25)  this  figure  will  be  the  linear  perspective  (27) 
of  that  contour,  and  hence  of  the  object  (32). 

35.  In  like  manner,  the  intersection  of  the  visual  ray  (23)  from 
any  one  point  of  the  given  object,  with  the  given  plane,  is  the 
perspective  of  the  point  from  which  that  ray  proceeded. 

36.  The  given  plane  is  therefore  called  the  perspective  plane  / 
and  is  understood  to  be  vertical,  unless  the  contrary  is  mentioned. 

Illustration. — In  Fig.  2,  let  E  be  the  position  of  the  eye,  ABC 
the  object,  as  a  wood  or  paper  triangle,  to  be  represented  ;  and  PQ, 
the  perspective  plane.  Then  AE,  BE,  and  CE,  represent  visual 
rays  from  the  corners  or  vertices  of  the  given  triangle.  Now  let 
«,  b,  and  c  represent  the  points  in  which  these  visual  rays  pierce 
the  perspective  plane  PQ,  then  dbc  will  be  the  perspective  of  ABC. 

37.  It  now  clearly  appears,  that,  in  order  to  find  the  perspective 
of  an  object,  three  things  must  be  given ;  the  object  itself,  the  posi- 
tion of  the  eye,  and  the  perspective  plane  (22).     Observe  here,  also, 
that,  as  lines  from  the  visible  points  of  the  object  to  the  eye  are 
visual  rays,  these  rays  become  known  as  soon  as  the  positions  of  the 
eye  and  of  the  object  are  given. 


DEFINITIONS    AND    GENERAL    PlilNdPLES. 


If  either  of  the  lines,  as  a#,  of  the  perspective,  were  prolonged 
either  way,  or  both  ways,  it  would  be  called  the  indefinite  perspec- 
tive of  the  original  line  as  AB. 

38.  Any  angle,  as  AEC,  formed  at  the  eye  by  two  visual  rays 


FIG.  2. 

is  called  a  visual  angle.  It  now  appears  that  any  line,  as  AC,  and 
its  perspective,  ac,  subtend  the  same  visual  angle.  The  reason, 
therefore,  ;why  an  object  and  its  perspective  present  the  same 
appearance  (25)  to  the  eye,  is,  that  they  subtend  the  same  visual 
angle;  for  the  apparent  size  of  any  object  depends  on  the  size  of  the 
visual  angle  which  includes  it.  Hence  if  two  equal  lines  be  in 
parallel  positions,  but  at  unequal  distances  from  the  eye,  the  further 
one  will  subtend  the  smaller  visual  angle  and  will  therefore  appear 
the  shorter  (18).  Also,  if  a  line  or  a  surface  be  viewed  obliquely, 
instead  of  directly,  it  will  appear  of  diminished  size,  and  is  said  to 
be  foreshortened. 

39.  The  position,  E,  of  the  eye,  and  the  form  and  position  of  the 
bject  ABC  remaining  fixed,  there  will  be  as  many  different  sizes 
and  forms  of  the  perspective,  «5c,  as  there  may  be  different  dis- 
tances and  positions  of  the  perspective  plane,  between  E  and  ABC 
And  these  various  forms  and  sizes  of  abc  will  all  be  true  perspec- 
tives of  ABC.  To  understand  this  completely,  it  is  only  necessary 
to  remember,  1°:  That  all  these  forms  of  abc  are  sections  of  the 
same  visual  pyramid  ABC — E  (34),  and  2° :  That  the  definition  of  a 
perspective  is  not,  a  figure  that  is  as  the.  original  object  appears; 
but,  only,  one  that  appears  as  that  object  does,  when  viewed  from 
the  sane  point  (25). 

2 


LINEAR   PERSPECTIVE. 


CHAPTER  H. 


THE  ELEMENTS    OF   PROJECTIONS. 

40.  It  is  evident  from  a  consideration  of  Fig.  2,  that  we  cannot, 
practically,  find  a  perspective  picture  directly  according  to  (35)  i.  e. 
directly  from  objects  themselves.     Visual  rays   are  invisible  and 
intangible,  and  we  cannot  conveniently  substitute  for  them,  threads 
from  every  point  of  an  object,  as  a  house,  to  a  fixed  point,  as  the 
top  of  a  stake,  taken  to  represent  the  place  of  the  eye,  and  then 
find  where  all  the  threads  pierce  a  paper  plane,  set-  up  between  the 
object  and  the  place  of  the  eye. 

41.  What  then  can  be  used  in  place  of  the  actual  object,  from 
which  to  make  its  perspective,  as  truly  as  if  found  mechanically,  as 
above  described  ?     We  employ  auxiliary  drawings,  which  show 
the  positions,  forms,  and  dimensions  of  the  original  objects,  just  as 
they  really  are,  and  from  such  drawings,  with  similar  representa- 
tions of  the  visual  rays,  we  construct  the  perspectives,  which  show 
those  objects  as  they  appear. 

42.  These  auxiliary  drawings,  which  show  the  given  object,  and 

its  visual  rays,  as  they  really 
are,  in  respect  to  form  and 
relative  position,  are  called 
projections.  To  the  expla- 
nation and  construction  of 
projections,  we  therefore 
turn,  as  the  next  thing  in  r 
order. 

43.  Illustration.  —  Let 
HH',  Fig.  3,  represent  a 
level  plane,  called  the  hori- 
zontal plane,  and  W  an 
upright  plane,  at  right  an- 
gles to  HH',  and  hence 
called  the  vertical  plane. 


FIG.  3. 


The  floor  and  any  wall  of  a  room,  would  be  such  a  horizontal  and 


THE    ELEMENTS    OF   PROJECTIONS.  19 


vertical  plane.     HV,  the  intersection  of  these  planes,  is  called  the 
ground  line. 

Next,  let  P  be  any  point  in  the  open  angular  space  between  these 
two  planes.  Then  let  Pp  be  a  straight  line  from  P,  perpendicular 
to  the  horizontal  plane,  HH',  and  meeting  it  at  some  point  repre- 
sented by  JP.  Likewise,  let  Ppr  be  a  line  from  P,  perpendicular  to 
the  vertical  plane,  VV,  and  meeting  it  at  p '.  Then  P  is  called 
the  horizontal  projection  of  P,  and  p'  is  the  vertical  projection  of  P. 

44.  Observe  now,  according  to  (41)  that  the  two  projections  of  a 
point  are  an  adequate  representative  of  the  real  position  of  the 
point.     For  p'q,  the  vertical  height  of  the  vertical  projection,^', 
above  the  ground  line  HV,  is  equal  to  the  real  height,  Pp,  of  the 
point  P  above  the  horizontal  plane.     Likewise  pq,  the  perpendicu- 
lar distance  of  the  horizontal  projection,^,  from  the  ground  line,  is 
equal  to  the  real  distance,  Pp',  of  the  point  in  space,  P,  from  the 
vertical  plane.     Hence  a  point  is  named  by  naming  its  projections  / 
thus,  we  describe  P  as  the  point  J0p'. 

45.  If  a  point,  as  S,  is  in  the  horizontal  plane,  it  coincides  with 
its  horizontal  projection,  s,  and  its  vertical  projection,  s',  must  be  in 
the  ground  line  HV.     Likewise,  if  a  point,  as  R,  lies  in  the  vertical 
plane,  it  is  its  own  vertical  projection,  r ',  and  its  horizontal  projec- 
tion, r,  must  be  in  the  ground  line. 

46.  By  considering  the  explanations  just  given,  we  are  led  to  the 
following  additional  practical  particulars.     First:  The  forms  of 
bodies  are  indicated  by  the  positions  of  the  points  which  compose 
or  limit  their  bounding  lines.     Hence,  if  the  distinguishing  points 
of  the  boundaries  of  an  object  be  projected  in  the  simple  manner 
just  explained,  and  if  the  projections  of  these  points  be  connected, 
in    each   plane,    the   projections   of   the    object    will   be    formed. 
Second:  p  represents  the  real  point,  P,  as  it  would  appear  if  seen 
from  above,  in  the  vertical  direction  Pp.    Likewise  p'  represents 
the  same  point  as  seen  in  looking  in  the  direction  Pp'.     Third:  In 
order  to  view  all  the  points  of  an  object  simultaneously  in  the  sam 
direction,  the  eye  must  be  at  an  indefinitely  great  distance  from  it 
Hence  projections  represent  objects  as  they  would  appear,  if  visible 
from  an  indefinitely  great  distance,  and  viewed  in  a  direction  per- 
pendicular to  each  plane  of  projection  in  succession. 

47.  Illustration.  Fig.  4.    GL  is  the  ground  line ;  GH,  the  hori- 
zontal plane  ;  and  GV,  the  vertical  plane.     ABC-D  is  a  triangular 
prism,  placed  with  its  edges  parallel  to  the  ground  line.     In  view- 
ing this  prism  from  a  great  distance  above  it,  so  as  to  look  at  all 
parts  at  once,  in  the  parallel  directions  Aa,  F/*,  <fcc.,  the  two  sloping 


20 


LINEAK   PERSPECTIVE. 


sides,  ACDF  and  ABDE,  will  be  visible.  Projecting  the  corners 
of  these  faces  by  vertical  projecting  lines,  Aa,  &c.,  as  in  Fig.  3,  we 
find  acfd  for  the  horizontal  projection  of  ACFD,  and  abde  for  the 
horizontal  projection  of  ABDE. 


FIG.  4. 

Likewise,  in  viewing  the  prism  in  the  direction  Aa',  its  front, 
ACDF,  only,  is  visible,  and  a' c' d'f  is  the  vertical  projection  of 
this  face. 

It  thus  appears  that  the  horizontal  projection  shows  the  true 
relative  distances  of  all  points  of  the  prism,  front  of  the  vertical 
plane  ;  and  that  the  vertical  projection  shows  the  true  heights  of  all 
points  of  the  prism  above  the  horizontal  plane.  That  is,  the  two 
projections,  together,  form  an  adequate  representative  of  the  form 
of  the  prism. 

48.  In  particular,  the  face  BCEF  is  parallel  to  the  horizontal 
plane,  and  beef,  its  horizontal  projection,  is  equal  to  it.     That  is, 
when  a  surface,  or  line,  is  parallel  to  a  plane  of  projection,  its  true 
size  is  shown  in  its  projection  upon  that  plane.     Again,  CB  is  a 
line  which  is  perpendicular  to  the  vertical  plane,  and  hence  the 
point  c'  is  its  vertical  projection.     That  is,  when  a  line  is  perpen- 
dicular to  either  plane  of  projection,  its  projection  on  that  plane  is 
a  point. 

49.  Continuing  to  examine  this  figure,  4,  with  reference,  now,  to 
its  lines  only,  it  appears  that  CF,  for  example,  is  parallel  to  both 
planes  of  projection,  and  hence  to  the  ground  line,  and  its  projec- 
tions cfaud.  c'f  are  both  parallel  to  the  ground  line,  and  each 


THE    ELEMENTS    OF    PROJECTIONS. 


21 


equal  to  CF.  That  is,  when  a  line  is  parallel  to  the  ground  line, 
each  of  its  projections  is  parallel  to  the  ground  line,  and  equal  to 

the  line. 

50.  Again :  AC,  for  example,  is  oblique  to  both  planes  of  pro- 
jection, but  it  is  in  a  plane  Aa'uc,  which  is  perpendicular  to  both  of 
these  planes,  and  both  of  its  projections,  ca  and  c'a',  are  perpen- 
dicular  to  the  ground  line,  and  each  is  less  than  the  line  AC.     Tha 
is,  when  any  line  is  oblique  to  both  planes  of  projection,  and  is  m 
a  plane  perpendicular  to  both,  each  of  its  projections  is  less  than 
the  line,  and  is  perpendicular  to  the  ground  line. 

51.  In  Fig.  5,  AB  is  a  line  which  is  paralled  to  the  horizontal 
plane  GH,  only,     a  b  and  a'b' 

are  its    projections.     ab=AB, 

and  a'b'  is  less  than  AB   and 

parallel    to    the     ground    line. 

That  is,  when  a  line  is  parallel 

to  the  horizontal  plane  only,  its 

horizontal  projection   is    equal 

and  parallel  to  itself,  and  its 

vertical  projection  is  parallel  to 

the  ground  line  and  less  than  FIG.  5. 

the  line  itself. 

52.  In  Fig.  6,  AB  is  a  line  parallel  only  to  the   vertical  plane 
VL,  ab,  its  horizontal  projection, 

is  parallel  to  GL,  and  less  than 
AB.  a'b ,  its  vertical  projection, 
is  equal  and  parallel  to  AB.  That 
is,  when  a  line  is  parallel  to  the 
vertical  plane  only,  its  vertical 
projection  is  equal  and  parallel  to 
the  line,  and  its  horizontal  projec- 
tion is  parallel  to  the  ground  line, 
and  less  than  the  given  line. 

53.  Finally,  in  Fig.  7,  AB  is  a 
line    which    is    oblique    to    both 
planes  of  projection.     Each  of  its 

projections,  ab  and  a'b',  is  less  than  AB,  and  oblique  to  GL. 
That  is,  when  a  line  is  oblique  to  both  planes  of  projection, 
both  of  its  projections  are  less  than  itself,  and  are  oblique  to  the 
ground  line.  Article  50  is  a  special  case  of  this  principle. 

Klines,  in  any  position,  are  parallel,  their  projections  will  be 
parallel. 


FIG.  6. 


22 


LINEAR  PERSPECTIVE. 


Lilies,  like  points,  are,  in  the  language  of  projections,  named  by 
naming  their  projections  (44).     Thus,  in  the  three  preceding  figures, 

the  line  itself,  AB,  would 
be  designated  as  the  line 
ab-a'b',  the  vertical  pro- 
jection being  distinguish- 
ed by  accents. 

54.  The  space  on  the 
same  side  of  the  vertical, 
or  perspective  plane,  as 
the  eye,  is  said  to  be  in 
front  of  it.  If,  now,  the 
horizontal  plane,  GH,  be 
extended  back  of  the  ver- 
tical plane  CV,  Fig.  4,  an 
angular  space  in  which  objects  may  be  placed,  will  be  formed 
behind  the  vertical  plane. 


Illustration.-!*  Fig.  8,  let  GH  be  the  front  part  of  the  horizon- 
tal plane,  and  GH'  its  back  part— the  vertical  plane,  GV,  being 
viewed  in   the   direction  of  the  arrows.     Let  ACbd  be  a  square 
prism  standing  on  the  back  part  of  the  horizontal  plane,  and  witl 
its  faces  parallel  and  perpendicular  to  the  vertical  plane, 
shown  in  the  figure,  C'D'c'cT,  a  rectangle,  equal  to  the  faceCI)^ 
will  be  the  vertical  projection  of  this  prism,  and  its  lower  base, 
abed,  will  constitute  its  horizontal  projection. 

Observe,  here,  that  when  a  body  stands  on  the  horizontal  plane, 
its  vertical  projection  will  stand  on  the  ground  line. 

Remarks.— a.  In  practice,  and  for  brevity,  the  horizontal  pro. 


THE    ELEMENTS    OF   PROJECTIONS.  23 

jection  is  called  the  plan,  and  the  vertical  projection,  the  eleva- 
tion. 

b.  If,  in  the  three  preceding  figures,  the  given  lines  had  been 
placed  behind  the  vertical  plane,  their  vertical  projections  would 
have  remained  the  same,  and  their  horizontal  projections  would 
have  appeared  behind  the  ground  line,  and  parallel  to  their  present 
positions.  The  student  is  recommended  to  reconstruct  these  figures 
accordingly. 

55.  In  (41)  projections  were  spoken  of  as  drawings  which  show 
objects  as  they  are,  rather  than  as  they  appear.  That  is,  projec- 
tions show  the  real  forms  and  dimensions  of  objects.  This  will 
now  appear  from  a  fuller  examination  of  Figs.  4  and  8.  In  Fig.  4, 
the  prism  being  placed  with  its  length  parallel  to  the  ground  line, 
and  one  of  its  rectangular  faces  parallel  to  the  horizontal  plane,  its 
horizontal  projection  gives  the  true  size  of  that  face ;  and  the  ver- 
tical projection,  the  altitude  perpendicular  to  the  faces  BCEF. 
Hence  the  prism  is  thus  fully  given  by  its  projections,  since  a  body 
is  said  to  be  completely  given,  when,  as  is  true  in  this  case,  such 
of  its  dimensions  are  known  as  enable  one  to  find  the  area  of  its  sur- 
face, or  its  solidity. 

Still  more  clearly  is  it  evident,  that,  in  Fig.  8,  the  plan,  abed, 
gives  the  true  width  and  thickness  of  the  prism,  and  the  elevation, 
the  width  and  height.  That  is,  the  two  projections,  together,  give 
the  three  dimensions  of  the  prism  in  their  real  size. 

Thus  it  will  be  seen  in  all  the  subsequent  figures,  that  the  pro- 
jections of  objects  give  their  real  forms,  as  seen  when  looking 
perpendicularly  towards  the  planes  of  projection,  and  that  when 
these  objects  are  placed  in  simple  positions  with  respect  to  those 
planes,  as  they  always  may  be,  their  projections  will  give  the  sim- 
ple dimensions  of  those  objects,  as  in  Fig.  8. 

Note. — The  figures  in  this  chapter,  and  others  like  them,  are  nothing  else  than 
examples  of  the  "  military  perspective,"  or  cabinet  projections,  explained  in  my 
"  Elementary  Projection  Drawing,"  Div.  IV.,  Chap.  V.  They  are,  therefore,  as  is 
evident  by  examination,  exact  constructions •,  representative  of  models  of  construe 
tions  in  space. 


24  LINEAR    PERSPECTIVE. 


CHAPTER  III. 

T1IE   CONSTRUCTION   OF  THE   PERSPECTIVES    OF    OBJECTS    FROM   THE II 

PROJECTIONS. 

56.  The  preceding  pictorial  representatives  of  models  of  projec- 
tions, may  suffice  to  render   the   subject    of  projections  of  given 
objects  intelligible.     We  therefore  proceed  to  illustrate  the  remain- 
ing points  in  (37)  viz.  the  projections  of  the  point  of  sight  (29),  the 
visual  rays  (30),  and  the  constructions  of  true  perspectives  of  objects 
from  their  projections,  instead  of  from  the  objects  themselves  (41). 

We  have  seen  that  the  eye  is  at  an  indefinitely  great  distance 
from  each  plane  of  projection,  successively,  in  viewing  objects  as 
seen  in  projection  (46).  But  objects,  as  seen  in  perspective,  are 
supposed  to  be  viewed  from  points  at  ordinary  finite  distances. 

The  point  of  sight  is  therefore  projected  like  any  other  point,  as 
in  Fig.  3. 

57.  Since  the  perspective  of  a  given  point  in  space,  is  the  point 
where  the  visual  ray  from  that  given  point  pierces  the  perspective 
plane   (35),  it  is  necessary,  in   the  next  place,  to  understand  the 
method  of  finding  the  point  in  which  a  given  line  pierces  any  verti- 
cal plane,  taken  as  a  perspective  plane  (36). 

For  this  purpose,  see  Fig.  9.  Here  GHH'  is  the  horizontal  plane, 
and  GV,  the  vertical  plane.  AE  is  any  line  in  space,  joining  the 
point  A,  behind  the  vertical  plane,  with  the  point  E  in  front  of  it. 
Aa  and  Ee  represent  the  vertical  projecting  lines  which  meet  the 
horizontal  plane  at  some  points  represented  by  a  and  e,  and  which 
therefore  determine  ae  as  the  horizontal  projection  of  AE.  Like- 
wise the  projecting  lines  Aa'  and  Ee',  which  are  perpendicular  to 
the  vertical  plane,  give  a'e'  as  the  vertical  projection  of  AE. 

This  being  established,  we  have,  by  referring  particularly  to  the 
line  AE  itself,  the  following,  as  the  first  method  of  explaining  the 
oonstruction  of  the  desired  point. 

It  is  evident  from  the  figure,  as  just  described,  that  any  line,  as 
AE,  must  pierce  the  vertical  plane  somewhere  in  its  own  vertical 
projection,  a'e'.  Also,  as  AE  is  directly  over  its  own  horizontal 
projection,  ae,  it  must  meet  the  vertical  plane  in  some  point  directly 


THE    CONSTRUCTION    OF   THE   PERSPECTIVE    OF    OBJECTS.  25 


FIG.  9. 

over  the  point  w,  where  its  horizontal  projection  meets  the  vertical 
plane,  in  the  ground  line.  Hence  AE  meets  the  vertical  plane  at 
ri,  the  intersection  of  a'e'  with  nri ,  a  line  perpendicular  to  the 
ground  line  at  n. 

58.  Now  suppose  the  line  itself,  AE,  to  be  removed,  leaving 
Only  its  projections,  ae  and  a'e,' ,  to  be  used  in  finding  the  point  n'. 
In  this  case,  we  have  the  following,  as  the  second  method — and  the 
usual   practical  one — of  explaining   the  construction  of  n'.     If  a 
point  lies  in  the  vertical  plane,  its  vertical  projection  is  the  point 
itself;    and   its  horizontal    projection  is   in  the   ground  line  (45). 
Conversely,  if  a  point  in  the  ground  line  is  the  horizontal  projec- 
tion of  some  point,  that  point  is  in  the  vertical  plane.     Hence  in 
Fig.  9,  n,  where  the  horizontal  projection,  ae,  of  the  given  line 
meets  the  ground  line,  is  the  horizontal  projection  of  that  point  of 
the  line  in  which  it  pierces  the  vertical  plane.     This  point  itself 
being,  as  already  explained,  in  the  vertical  projection,  a'e' ,  of  the 
given  line,  and  also  in  a  perpendicular  to  the  ground  line  at  n,  it  is 
at  ri,  the  intersection  of  nn'  with  a'e'. 

Remarks. — a.  The  construction  of  n'  being  of  constant  occur- 
rence in  exact  perspective  drawings,  both  of  the  above  explana- 
tions should  be  memorized,  as  well  as  clearly  understood,  until 
they  become  thoroughly  familiar. 

b.  It  is  now  evident  that  if  E  is  the  position  of  the  eye,  and 
GLV  the  perspective  plane,  AE  is  a  visual  ray,  and  n'  is  the  per- 
spective of  the  point  A,  as  seen  by  the  eye  at  E. 

59.  In  further  illustration  of  the  manner  of  finding  the  perspeo 


26 


LINEAR   PERSPECTIVE. 


tives  of  objects  from  their  projections,  Fig.  10  is  added,  which  is  a 
pictorial  representation  of  the  construction  of  the  perspective  of  a 
straight  line  in  space. 


FIG.  10. 

GHFP  is  the  horizontal  plane.  GV,  the  vertical  plane,  is,  as 
usual,  taken  also  for  the  perspective  plane.  Let  AB  be  any 
oblique  line,  behind  the  perspective  plane,  and  meeting  the  horizon- 
tal plane  at  B,  and  whose  perspective  is  to  be  found.  Let  E,  in 
front  of  the  perspective  plane,  be  the  point  of  sight.  By  (43)  e 
and  e'  are  the  projections  of  the  point  of  sight.  Likewise,  ar 
and  a'  are  the  projections  of  A.  The  point  B,  being  in  the  hori- 
zontal plane,  is  its  own  projection  on  that  plane,  and  by  (45)  5',  in 
the  ground  line,  is  its  vertical  projection.  Therefore,  $B  and  a'b' 
are  the  projections  of  the  given  line.  Now,  AE  is  the  visual  ray 
from  A,  the  upper  end  of  the  line  AB ;  ae  and  a'e'  are  the  projec- 
tions of  this  ray.  Then  by  (58)  this  ray  pierces  the  perspective 
plane  at  A',  which  is,  therefore,  (585)  the  perspective  of  A.  Like 
wis-e  BE  is  the  visual  ray  from  B,  the  foot  of  the  given  line  ;  and 
Be  and  b'e'  are  its  projections.  This  ray  pierces  the  perspective 
plane  at  B',  which  is,  therefore,  the  perspective  of  B.  Hence  A'B' 
is  the  perspective  of  AB. 

Remarks. — a.  Since  the  perspective,  or  vertical  plane,  is  placed 
between  the  eye  and  the  given  object,  the  object  lies  behind  the 
perspective;  plane,  as  in  Figs.  8  and  10;  hence  its  plan  will  appear 
behind  the  ground  line. 


THE    CONSTRUCTION    OF    THE    PERSPECTIVE    OF    OBJECTS.  27 

b.  In  all  cases  where  two  planes  of  projection  are  used,  as  just 
shown,  the  vertical  plane  of  projection  is  also  the  perspective  plane, 
and,  therefore,  contains  both  the  vertical  projection  and  the  per- 
spective of  the  given  object. 

c.  By  erasing  the  lines  AB,  BE,  and  AE,  so  as  to  leave  only  their 
projections,  also  the  projecting  lines  Aa,  Aa',  B#',  Ee,  and  E#',  the 
remaining  lines  would  show  pictorially  the  construction  of  the  per 
spective,  A'B',  from  the  projections,  only,  of  AB,  and  of  the  ray 
AE. 

The  construction  of  such  a  figure  is  left  for  the  student  to  make, 


28 


LINIilAR    PERSPECTIVE. 


CHAPTER  IV. 

KEAL   PROJECTIONS,   AND   PERSPECTIVES    MADE   FROM  THEM. 

60.  All  the  preceding  figures  are  only  the  pictures  of  projec- 
tions, and  pictures  of  the  perspectives,  made  from  those  projections, 
and  not  the  projections  and  the  perspectives  themselves.  They  are 
pictorial  representatives  of  the  models  which  would  show  given 
objects,  the  eye,  the  perspective  plane,  and  visual  rays,  as  they 
actually  exist  in  space. 

It  is,  therefore,  next  to  be  found  how  these  projections  and  per- 
spectives themselves  are  represented.  In  doing  this,  we  seek  first 
the  method  of  representing  the  planes  of  projection,  which  are 
really  at  right  angles  to  each  other,  upon  a  single  flat  surface,  as  a 
sheet  of  paper. 


FIG.  11. 

61.  From  Fig.  11  it  is  evident  that,  if  the  vertical  plane,  GWn 
l»e  revolved  directly  back  about  the  ground  line,  GL,  as  an  axis, 
until    it  coincides  with  the  back  portion,  GH',  of  the  horizontal 
plane,  all  the  points  of  the  vertical  plane  wTill  describe  circular  arcs 
in  the  direction  of  the  arrows,  and  in  planes  perpendicular  to  the 
ground  line.     If,  then,  A  be  a  point  in  space,  and  in  front  of  the 
vertical  plane,  and  a  and  a\  its  projections,  a\  will  describe  the 
quadrant  a\  a',  and  will  be  found  in  the  line  ana',  a  perpendicular 
to  the  ground  line  through  the  horizontal  projection,  a,  of  the 
given  point. 

62.  According,  now,  to  this  illustration,  the  following  method 


HEAL    PROJECTIONS,    AND    PERSPECTIVES    MADE    FROM    THEM.  29 

is  universally  agreed  upon  as  the  one  to  be  practically  adoj  ted  in 
representing  projections.  m 


G 


FIG.  12. 

Having  drawn  any  line,  GL,  Fig.  12,  upon  the  drawing  paper, 
to  represent  the  ground  line,  it  is  understood  that  all  that  portion 
of  the  paper  below  or  in  front  of  the  ground  line,  represents  the 
front  part  of  the  horizontal  plane  of  projection,  and  that  the  por- 
tion above  or  back  of  the  ground  line,  represents  both  the  back 
part  of  the  horizontal  plane,  GH',  Fig.  11,  and  the  vertical  plane, 
GY,  Fig.  XL 

Hence,  make  na  and  na ',  Fig.  12,  equal  respectively  to  na  and 
na\,  or  na'^  on  Fig.  11,  and  a  and  a'  will  represent,  not  the  pic- 
tures of  the  projections  of  A,  but  the  projections  themselves  of 
that  point.  Accordingly,  as  in  (44),  the  point  supposed  is  named 
by  its  projections,  and  we  say  "  the  point  ««',"  meaning  the  point 
in  space  whose  projections  are  a  and  a',  and  which  is  at  the  height, 
a'n,  above  the  horizontal  plane,  and  distance,  an,  in  front  of  the 
vertical  plane. 

63.  Having  now  shown,  pictorially,  the  manner  of  representing 
the  projections  of  points,  upon   the  planes  of  projection  when  in 

heir  real  position  (43)  and  as  shown  after  revolution  (61)  and  the 
manner  of  finding  the  perspective  of  a  point,  or  a  line  (58^-59)  the 
way  is  prepared  for  the  connected  review  embraced  in  the  three 
following  figures,  which  show  first :  a  pictorial  representation  of 
the  construction  of  the  perspective  of  a  point,  on  the  perspective 
plane  in  its  real  position;  second:  a  similar  view  after  the  revolu- 
tion of  the  perspective  plane  into  the  plane  of  the  paper  ;  and 
third:  an  actual  construction  of  the  real  perspective  of  a  point. 

64.  In  Fig.  13,  E  is  the  position  of  the  eye,  e  is  its  horizontal 
projection,  and  e'  is  its  vertical  projection.     Ce'  =  Ee  shows  the 


80 


LINEAR   PERSPECTIVE. 


height  of  the  eye  above  the  horizontal  plane,  and  C  e  =  E  e'  shows 
its  distance  in  front  of  the  vertical  plane,  LGP,  which  is  also  the 
perspective  plane. 


FIG.  13. 

A  is  the  point,  whose  perspective  is  to  be  found,  a  is  its  horizon- 
tal projection,  and  a'  its  vertical  projection.     Ba'  =  Aa    shows 
the  height   of  A  above   the  horizontal   plane  GHH;  aB    =  a' A 
shows  its   distance    behind  the   perspective  plane  LGP,  and   CB 
shows  its  distance  to  the  right  of  the  observer  standing  at  e.     Thug 
the  position   of   the  point   is  completely  indicated  with   respect, 
both  to  the  eye  at  E,  and  the  perspective  plane  (22). 
We  have,  now,  the  following  statement  of  this  problem. 
Given:  GHH, 
LGP, 
E,  and 
A. 

Required,  the  perspective  of  A,  given  in  its  projections  a  and 
a'/  on  LGP,  as  seen  from  E,  given  also  by  its 
projections  e  and  e'. 

Construction.  Draw  ae  and  afe\  and  note  n.  At  n  erect 
ns  perpendicular  to  GL  and  note  s,  where  it 
meets  a'e'.  Then  s  will  be  the  required  per- 
spective  of  aa',  that  is,  of  A  (58). 

65.  This  being  established,  we  proceed  according  to  (63)  to 
represent,  pictorially,  the  revolution  of  the  perspective  plane  back- 
ward into  the  horizontal  plane,  so  as  to  show  the  above  construc- 
tion in  a  single  plane,  as  is  done  in  practice.  The  several  quarter 


REAL   PROJECTIONS,    AND    PERSPECTIVES   MADE   FROM   THEM.          31 

circles,  Fig.  13,  represent  the  revolutions  of  the  several  points  a',  e\ 
etc.,  about  GL  as  an  axis,  till  they  reach  the  horizontal  plane. 

Thus  LGP'  is  the  revolved  position  of  the  perspective  plane 
LGP,  and  a",  s',  and  e"  are  the  revolved  positions  of  a',  s,  and  e' 
respectively.  Then  a"  e"  is  the  revolved  position  of  the  vertical 
projection  a'  er  of  the  ray  AE,  and  n  s'  the  revolved  position  of 
rts,  giving  s',  as  the  revolved  perspective. 

Since  the  horizontal  plane  is  unmoved,  all  points  upon  it  remaii 
fixed,  and  a  and  a"  are  the  projections  of  A,  e  and  e"  the  projec- 
tions of  E,  and  ae  and  a"e"  the  projections  of  the  visual  ray  AE. 

Observe,  now,  that  when  the  two  planes  are  no  longer  shown  in 
their  real  position,  the  ray  itself,  AE,  can  no  longer  be  shown,  so 
that  Fig.  14  shows  separately,  and  pictorially,  all  of  Fig.  13  that 
can  appear  when  the  two  planes  are  represented  as  one  surface. 
Fig.  14  evidently  shows,  however,  according  to  the  last  article,  all 


&  p 

FIG.  14. 

that  is  essential  in  finding  s,  the  perspective  of  aar  as  seen  from 
the  eye  at  eer. 

67.  Finally ;  Fig.  15  shows  the  pictorial  representation  in  Fig. 
14,  transformed  into  an  actual  construction,  according  to  (62). 

[Note. — While  a  single  complete  illustration,  fully  explained, 
suffices  for  the  purposes  of  a  text  book,  the  learner,  in  order  to 
avoid  confusion  of  mind  in  his  progress,  should  make  himself  per- 
fectly familiar  with  each  successive  stage  of  the  subject,  by  con- 
structing a  variety  of  figures,  similar  to  those  thus  far  given.] 

.  Proceeding,  now,  with  Fig.  15,  aa'  is  the  given  point,  at  a  distance 
equal  to  ad,  back  of  the  vertical  plane,  and  at  a  distance  equal  to 
ad  above  the  horizontal  plane,  ee'  is  the  point  of  sight,  at  a  dis- 
tance equal  to  eb  in  front  of  the  vertical  plane,  and  at  a  distance 
equal  to  e'b,  above  the  horizontal  plane. 

From  the  preceding  descriptions,  it  follows  that  ae  is  the  horizon- 
tal projection  of  a  visual  ray  from  the  given  point  aa' ,  and  a'e'  is 


32 


LINEAR   PERSPECTIVE. 


its  vortical  projection.  Then  p,  the  point  where  the  horizontal 
projection  of  this  ray  meets  the  ground  line  GL,  is  the  horizontal 
projection  of  that  point  of  the  ray  in  which  it  pierces  the  vertical, 


Cr 


fl 


FIG.  15. 

^.  e.  the  perspective  plane  (58).  The  latter  point  being  also  in  the 
vertical  projection  of  the  ray  (57,58)  is  at  P,  the  intersection  of  a'e' 
with  joP,  a  perpendicular  to  the  ground  line  at  p.  Therefore  P  is 
the  perspective  of  aa',  as  seen  from  the  point  ee' '. 

68.  In  order  to  find  the  perspective  of  any  object,  we  have  only 
to  find  the  perspectives  of  its  separate  points,  exactly  as  just 
described.  Hence  the  following  explanations  will  not  be  so  minute 
for  each  point,  as  the  one  just  given.  We  shall  now  proceed  to 
explain  the  construction  of  the  perspectives  of  the  leading  elemen- 
tary solids,  viz.  the  prism, pyramid,  cylinder,  cone,  and  sphere;  toge- 
ther with  various  subordinate  practical  particulars.  And  though  this 
may  not  in  itself  interest  the  learner  as  much  as  the  representation 
of  objects  whose  perspectives  have  more  of  pictorial  effect,  yet  the 
recollection  that  no  other  method  will  so  jconcisely  afford  an  equally 
abundant  variety  of  universally  useful  methods  of  practical  ope^ 
ration,  in  subsequent  practical  examples,  may  suffice  to  com- 
pensate for  the  comparative  inelegance  of  the  perspectives  now  to 
be  explained. 

EXAMPLE  1. — To  find  the  Perspective  of  a  Vertical  Square 
Prism,  situated  as  shown  in  Fig.  8. 

From  Fig.  8  it  is  evident,  that  when  the  perspective  plane  is 
revolved  backward,  the  lines  c'.C'  and  dT)'  of  the  elevation  (55a)  will 
exactly  fall  upon  the  lines  cb  and  da  of  the  plan.  Hence,  in  Fig.  16, 
AJBCD  and  c'd'  C'D'  are  the  correct  projections  of  a  square  prism 


KEAL   PROJECTIONS,    AXD    PERSPECTIVES    MADE    FROM   THEM. 


33 


standing  on  the  horizontal  plane,  at  the  distance  d'D  behind  the 
vertical  or  perspective  plane. 


FIG.  16. 


Let  EE'  be  the  position  of  the  eye.  The  rays  from  c'  and  C', 
being  one  directly  over  the  other,  have  the  same  horizontal  projec- 
tion, CE.  Then  by  (67)  CE  —  c'E'  is  the  visual  ray  (585)  from  the 
front  left  hand  corner,  Cc',  of  the  baseband/  is  the  perspective 
of  that  corner.  CE  —  C'E'  is  the  ray  from  the  corresponding  upper 
corner  CC'.  It  pierces  the  perspective  plane  at  F,  which  is  there- 
fore the  perspective  of  CC'.  Likewise  the  visual  ray  DE  —  e?'E' 
pierces  the  perspective  plane  at  o,  which  is  therefore  the  perspec- 
tive of  the  point  DC?';  and  the  visual  ray  DE  —  D'E'  pierces  the 
perspective  plane  at  O,  giving  the  perspective  of  the  point  DD'. 
Hence' the  figure  F/Oo  is  the  perspective  of  CD  —  c'd'  C'D',  the 
front  face  of  the  given  prism.  Finally,  BE  and  c?'E'  are  the  projec- 
tions of  the  visual  ray  from  the  right  hand  back  corner,  B,  d',  of  the 
base  of  the  prism.  This  ray  pierces  the  perspective  plane  at  n,  the 
perspective  of  this  corner.  Also  BE — D'E'  is  the  visual  ray  from 
the  corresponding  corner,  B,D',  of  the  upper  base.  This  ray  gives 
N  as  the  perspective  of  B,D'.  Then  drawing. ON",  on,  and  riN,  we 
shall  have  the  complete  perspective  of  the  visible  edges  of  the 
given  prism. 

69.  Remarks. — a.  Observe  tliat/b  and  FO  are,  by  the  construc- 
tion, parallel  to  the  ground  line  GL,  and  to  the  lines  CD  —  c'd',  and 

3 


34 


LINEAR   PERSPECTIVE. 


CD  -  C'D',  of  which  they  are  the  perspectives.  Also  that  the  right 
hand  edges,  BD  —  d'  and  BD  —  D',  of  the  bases,  are  perpendicular 
to  the  perspective  plane,  while  their  perspectives  on  and  ON  meet 
at  E',  if  produced.  This  is  easily  explained  from  elementary  geo- 
metry. Planes  containing  the  eye  and  the  vertical  edges,  as  at  C 
and  D,  are  vertical  planes,  standing  on  the  lines  CE  and  DE. 
Hence  they  must  intersect  the  perspective  plane,  which  is  also  ver- 
tical, in  lines,  &F  and  AO,  which  will  be  vertical,  that  is  parallel  to 
the  vertical  edges  of  the  prism,  at  C  and  D. 

In  like  manner,  since  the  top  and  bottom  edges,  c'd'  and  C'D', 
are  parallel  to  the  perspective  plane,  the  planes  passed  through 
them  and  the  eye,  must  intersect  the  perspective  plane,  in  lines  fo 
and  FO,  which  will  be  parallel  to  those  given  edges.  And,  gen- 
erally, the  perspectives  of  all  parallels  to  the  perspective  plane,  are 
parallel  to  the  lines  themselves. 

1.  Planes  through  the  eye,  and  the  edges  DB-cf  and  DB-D' 


<t 


FIG.  17. 

which  are  perpendicular  to  the  perspective  plane,  will  contain  tne 
visual  rays  from  D,<f  and  DD'  and  will,  themselves,  be  perpendicu- 


REAL    PROJECTIONS,    AND    PERSPECTIVES    MADE    FROM    THEM.          35 

lar  to  that  plane.  They  will  therefore  intersect  each  other  in  a  line, 
perpendicular  to  the  perspective  plane  at  E'.  Hence  their  inter- 
sections with  the  perspective  plane,  which  will  be  the  indefinite 
perspectives  (37)  of  the  edges  contained  in  them,  will  be  the  lines 
d'E',  and  D'E'. 

EXAMPLE  2. — To  find  the  Perspective  of  a  Triangular 
Pyramid, 

This  example  embraces  the  perspectives  of  oblique  lines,  abc — v 
and  a'b'c'  —  v'^  Fig.  17,  are  the  projections  of  a  triangular  pyramid^ 
standing  on  the  horizontal  plane,  and  behind  the  vertical  plane.  E 
and  E'  are  the  projections  of  the  point  of  sight.  aE —  #'E'  is  the 
visual  ray  from  the  point  aar  of  the  base  of  the  pyramid.  This 
ray  pierces  the  perspective  plane  at  A,  the  intersection  of /A  and 
a'E'.  A  is  therefore  the  perspective  of  aa '.  Likewise,  by  the  rays 
£E  —  &'E',  and  cE  —  c'E',  we  find  B  and  C,  the  perspectives  of  bbf 
and  cc'.  vE  —  v'~E>'  is  the  visual  ray  from  the  vertex  vv',  and  it 
pierces  the  perspective  plane  at  V,  the  intersection  of  AY  and 
t/E',  giving  V  as  the  perspective  of  vvf. 

Joining  the  points  now  found,  ABC  — V  is  the  perspective  of 
the  given  pyramid  abc-v  —  a'b'c'  -v',  as  seen  from  EE'. 

Remark. — The  student  should  construct  other  figures  by  the 
above  method,  till  quite  familiar  with  it. 


LINEAR    PERSPECTIVE. 


CHAPTER  V. 

BtiMOVAL    OF    PEACTICAL     DIFFICULTIES    ARISING     FROM    THE   CONFCJ 
SIGN    OF   PROJECTIONS    AND   PERSPECTIVES. 


§  I.  — First  Method. 


Translation,  forward,  of  the  Perspective, 
Plane. 


70.  The  perspective  plane  being  between  the  eye  and  the  given 
object,  the  plan  of  that  object  must   lie   behind  the  ground  line. 
Also,  as  the  perspective  plane  contains  both  the  vertical  projection 
and  the  perspective  of  the  object,  these  two  must  both  fall  upon  the 
plan,  when  the  perspective  plane  is  revolved  back  into  the  horizon- 
tal plane  ;  as  seen  in  the  last  two  examples. 

The  confusion  of  lines  arising  from  this  source  is  sufficiently  ap- 
parent from  Figs.  16  and  17,  though  they  embrace  very  simple 
objects,  and  remove  the  perspectives  as  far  as  possible  from  the 
projections,  by  placing  the  eye  considerably  to  one  side  of  the 
projections. 

71.  Hence,  before  proceeding  further  with  practical  constructions, 
we  shall  present  a  simple  method  of  obviating  the  difficulty  just 
mentioned.     This  method  consists  in  transferring  the  perspective 
plane,  with  all  the  points  in  it,  directly  forward,  far  enough  to  allow 
it  to  be  revolved  back  so  as  to  lodge  the  figures  in  it  entirely  below, 
or  in  front  of,  the  plan. 


Ar 


P' 


a  :£' 


-*r 

FIG.  18. 


-E 


This  method  is  illustrated  in  Fig.  18.  A  is  a  point  whose  projec- 
tions are  a  and  a',  on  planes  seen  edgewise  and  in  their  real  posi- 
tions at  right  angles  to  each  other,  at  aGG'  and  GP.  E  is  the 
place  of  the  eye.  Then  X  represents  die  t»«rsi>e<»,tive  of  aa'. 


REMOVAL    OF    PRACTICAL   DIFFICULTIES.  37 

When,  now,  the  perspective  plane  GP  is  revolved  back  as  shown 
by  the  arrows,  carrying  a'  and  X  to  a,"  and  X',  a,X'  and  a"  will  be 
crowded  together.  But  suppose  the  perspective  plane  to  be  first 
moved  forward — carrying  along  the  points  a'  and  X — to  a  new 
position  G'P',  and  then  to  be  revolved.  The  perspective,  X",  will 
then  appear  at  X'",  free  from  the  plan  ;  and  it  may  also  be  freed 
from  the  elevation,  in  practice,  by  erasing  portions  of  the  latter 
from  time  to  time,  as  the  construction  of  the  perspective  progresses, 
or  by  transferring  only  the  perspective  points. 

The  elementary  examples  of  the  last  chap  or  are  here  continued, 
according  to  the  method  just  explained. 

EXAMPLE  3.— To  find  the  Perspective  of  a  Cube,  which 
stands  obliquely  with  respect  to  the  perspective  plane. 

See  Fig.  19.  aceg  is  the  plan  of  a  cube  thus  situated,  and  a'Vc'f 
is  its  elevation. 

The  ground  line  GL  indicates  the  first  position  of  the  perspective 
plane,  and  G'L/  shows  its  position  after  translation  forward.  E  is 
the  horizontal  projection  of  the  point  of  sight.  Being  in  the  hori- 
zontal plane,  its  position  is  not  affected  by  the  translation  of  the 
perspective  plane.  E'  is  the  vertical  projection  of  the  point  of 
sight,  shown  only  on  the  second  position  of  the  perspective  plane, 
since  it  is  used  only  there.  For  a  similar  reason,  the  vertical  pro- 
jections of  the  visual  rays  are  shown  only  on  the  second  position 
of  the  perspective  plane.  #E  is  the  horizontal  projection  of  the 
visual  rays  from  the  two  points  oaf  and  afi  (Ex.  1.).  By  making 
b"a"=b'a',  and  in  a!l>'  produced,  we  find  the  projections  of  of  and 
V  upon  the  second  position  of  the  perspective  plane.  Likewise 
we  find/",  e",  c",  etc.  Then,  for  example,  «E  and  a"E'  are  the 
projections,  employed,  of  the  visual  ray  from  a, a" ;  or,  more 
briefly  (58)  aE — a"E'  is  the  visual  ray  from  a,a".  This  ray 
pierces  the  perspective  plane  at  A,  the  intersection  of  a"E'  with 
the  perpendicular  to  GL,  at  A,  where  the  horizontal  projection, 
aE,  of  the  ray  meets  the  real,  that  is  the  original  position  of  the 
ground  line  (57-8).  Then  A  is  the  required  perspective  of  a,aff. 
Other  points  as  B,F,  etc.,  of  the  perspective  of  the  cube  may  be 
found  in  a  precisely  similar  manner.  The  construction  of  some  of 
the  points  is  therefore  omitted,  to  avoid  unnecessary  confusion  of 
the  figure.  Thus,  the  perspective  of  the  point,  c,c?"  will  be  at  the 
intersection  of  a  line  c?"E'  with  tLe  perpendicular  to  GL  at  n.  The 
perspective  of  the  back  upper  corner  g,g"  is  likewise  at  the  inter- 


38 


LINEAR   PERSPECTIVE. 


section  of  a  ray  from  g"  to   2?',  with  the  perpendicular  to  GL 
at  o. 

To  avoid  the  acute  intersections,  as  at  B,  by  the  method  of  two 
planes,  without  setting  E,E'  far  to  one  side,  as  in  Fig.  1  7,  trans- 


late 


as 


the  points, 
aj)",  of  the  given 
object,  only,  to  one 
side  in  a  direction 
parallel  to  the 
ground  line,  and 
then  find  their  per- 
spectives, as  B'  (not 
shown)  which  will 
be  well  defined. 
Then  a  parallel  to 
the  ground  line, 
through  B',  will  in- 
tersect either  £"E', 
or  AB,  giving  B  by 
a  well  defined  in- 
tersection. Observe 
that  E,E'  is  not 
moved. 

Remark.  —  The 
perspectives  of  other 
plane-sided  objects, 
in  various  positions, 
should  be  construct- 
ed by  the  learner, 
by  the  method  just 
explained.  For  ex- 
ample, let  Fig.  17 
be  re-constructed 
according  to  the 
method  of  Fig.  19. 


§  II.     Second  Method.     Use  of  three  Planes. 

72.  The  confusion  of  the  diagrams,  arising  from  the  confounding 
together  of  the  perspective  with  either  or  both  of  the  projections 


REMOVAL    OF   PRACTICAL    DIFFICULTIES.  39 

of  the  given  object,  may  be  still  further  avoided  by  making  the 
perspective  plane  a  third  plane,  separate  from  both  of  .the  planes 
of  projection,  and  at  right  angles  to  both  of  them. 
This  is  accomplished  in  the  manner  illustrated  in  Fig.  20. 


FIG.  20. 

OHH  is  the  horizontal  plane  of  projection ;  VV,  the  vertical 
plane  of  projection,  and  OLQ  the  perspective  plane.  P  is  a  point 
in  space,  whose  perspective  is  to  be  found,  p  represents  its  hori- 
zontal, and p'  its  vertical  projection.  E  is  the  position  of  the  eye, 
e  its  horizontal,  and  e'  its  vertical  projection.  Then  PE  represents 
the  visual  ray,  whose  intersection  with  OLQ  will  be  the  perspec- 
tive of  P.  pe  is  the  horizontal,  and  p'  e'  the  vertical  projection 
of  this  ray.  The  perspective  plane  OLQ  is  perpendicular  to  both 
of  the  other  given  planes,  and  LQ  is  its  intersection  with  the  ver- 
tical plane  of  projection.  LQ  is  called  the  trace  of  OLQ  upon  the 
vertical  plane  of  projection.  Then,  as  in  previous  cases,  Pi,  the 
perspective  of  P,  is  in  the  line  n  Pl5  perpendicular  to  the  ground 
line  OL  at  n.  Likewise  it  is  obviously  in  the  line  r  Pj,  perpendicu- 
lar to  the  trace  LQ  at  r.  Hence  P  is  at  the  intersection  of  n  Pi 
and  r  P,. 

73.  Now  in  order  to  bring  all  three  of  these  planes  into  a  single 
surface,  as  is  done  in  practical  drawing,  the  perspective  plane  may 
be  revolved  about  its  trace  LQ  till  it  coincides  with  the  vertical 
plane  W,  which  may  then  be  revolved  back  as  usual  around  t-Le 
principal  ground  line,  HLj.  But  by  such  a  proceeding,  the  per- 
spective of  an  object  would  by  revolution  fall  upon  the  vertical 


40 


LINEAR   PERSPECTIVE. 


projection  of  that  object.  Hence  the  perspective  plane  is  moved 
towards  the  eye,  and  parallel  to  its  first  position  to  some  con- 
venient  new  position  as  n^r^  before  being  revolved.  Then,  as 
every  point  of  the  perspective  plane  moves  parallel  to  the  ground 
line,  n  will  appear  at  n^  and  r  at  /•„  and  after  revolution  in  the  di- 
rection n\n^  the  vertical  line  ?iP}  will  appear  at  n^P^  and  th 
horizontal  linerP,,  at  rfz.  Hence  P  will  be  the  perspective  of 
P,  after  the  translation  and  first  revolution  of  the  perspective 
plane. 

74.  The  perspective  of  a  point  by  the  method  of  three  planes, 


\  .--•*£ 


u. 


FIG.  21. 

shown  pictorially  in  Fig.  20,  is  shown  as  an  actual  construction  in 
Fig.  21.  The  former  figure  is  exactly  transformed  into  the  latter 
by  making  the  corresponding  distances  equal  in  both,  and  by  letter- 
ing the  same  points  with  the  same  letters,  so  far  as  shown  at  all. 

pp  is  the  given  point,  given  by  its  projections,  ee  is  likewise 
the  point  of  sight,  9iLr  the  first,  and  n^L^  the  second  position  of 
the  perspective  plane,  thus  indicated  as  at  right  angles  to  both 
planes  of  projection,  pe—p'e'  is  the  visual  ray  from  pp',  which 
pierces  the  perspective  plane  nLr  at  a  point  whose  projections  are 
n  and  r.  After  translating  this  plane,  parallel  to  the  ground  line, 
to  the  position  n^L[r^  these  points  appear  at  n1  and  rt.  Then,  by 
revolving  the  perspective  plane  from  n{L\r\  into  the  vertical  plane 
of  projection,  the  point  n\r\  describes  a  horizontal  arc  about  the 
point  L,,  TI  as  a  centre.  The  projections  of  this  arc  are  n^  and 
»*,P ,  and  P2  thus  appears  as  the  perspective  of  pp'. 


REMOVAL    OF    PllACTICAL   DIFFICULTIES. 


41 


Remarks. — a.  The  perspective  plane  must,  in  Fig.  21,  be  trans- 
lated to  the  right  so  as  to  revolve  to  the  left,  in  order  that  the  right 
hand  of  the  perspective  may  continue  to  correspond  with  the  right 
hand  of  the  object  drawn.  This  will  be  obvious  on  inspection  in 
the  succeeding  examples,  wherever  three  planes  shall  be  used. 

b.  Either  of  the  methods  of  disposing  of  the  perspective  plane, 
explained  in  this  chapter,  will  be  used  at  pleasure  in  the  solutions 
which  follow.  The  student  is  advised  to  solve  the  subsequent 
problems,  on  three  planes,  when  two  are  used  by  the  author,  and 
vice  versa. 

To  assist  therefore  in  becoming  more  familiar  with  the  use  of 
three  planes,  the  following  practical  problem  is  giveo. 


Fia.  22. 

EXAMPLE  4. — To  find  the  Perspective  of  an  Obelisk,  com- 
posed of  a  frustum  of  a  long  pyramid,  capped  oy  a  snort 
pyramid. 


42  LINEAR   PERSPECTIVE. 

Let  the  square  acbd—c'd'.  Fig.  22,  be  the  horizontal  and  verti- 
cal projections  of  the  base  of  the  obelisk  ;  and  v—fnot — v'-n't'  the 
projections  of  the  cap  pyramid. 

Let  PQP'  be  the  first  and  real  position  of  the  perspective  plane, 

at  right  angles  to  botli  planes  of  projection.     Let  PjQiP'j,  be  its 

second  position,   parallel  to  the  first,  from  which  it  is  revolved 

round  P'iQh  its  intersection  with  the  vertical  plane,  until  it  coin 

cides  with  that  plane.     EE'  is  the  point  of  sight. 

To  find  the  perspective  of  any  point,  as  aa',  of  the  base.  «E  and 
a'E'  are  the  projections  of  the  visual  ray  from  this  point.  This  ray 
pierces  the  perspective  plane  at  gg'.  This  point,  after  translation, 
appears  at  g\.g^  found  by  drawing  gg^  and  g'gl,  parallel  to  the 
ground  line.  After  its  revolution  through  the  horizontal  quarter 
circle  whose  projections  are  g^  and  g^  A,  it  appears  at  A,  the  inter- 
section of  ffi'A  with  <72A,  perpendicular  to  the  ground  line  QQi. 

In  like  manner  C  and  B,  the  perspectives  of  cc'  and  lib'  are 
found. 

Note  that  W,  the  invisible  corner  of  the  base  as  seen  in  ver- 
tical projection,  is  the  right  hand  corner,  to  the  eye  at  EE'  looking 
in  the  direction  Ew. 

To  find  the  perspective  of  any  point  of  the  cap  pyramid,  we  also 
proceed  just  as  before.  Thus,  oE  —  o'E'  is  the  visual  ray  from  the 
corner  oo'.  This  ray  pierces  the  perspective  plane  PQP'  at  pp \ 
which  is  translated  to  £>$>/,  and  from  that  position  revolved  in  a 
horizontal  arc,  as  before,  to  O,  the  perspective  of  oo'. 

Remarks. — a.  Every  point  of  the  perspective  being  thus  found 
in  precisely  the  same  manner,  the  construction  of  several  of  them 
is  left  to  be  made  by  the  student. 

b.  Observe  also,  that  as  the  operations  in  Figs.   21   and  22  are 
precisely  similar,  the  perspective  of  any  object,  by  the  method  of 
three  planes,  is  simply,  and  only,  a  continued  repetition  of  the  con. 
Btruction  of  the  perspective  of  a  single  point,  as  in  Fig.  21. 

c.  Practice  is  required,  however,  to  enable  the  learner  to  under 
stand  readily  tine  form  and  position  of  any  given  object  from  it* 
projections^  and  to  determine  easily,  by  mere  inspection,  the  pro- 
jections of  those  points  which  are  seen  from  the  given  point  of 
sight.     Hence,  again,  the  student  is  advised  to  construct  the  per- 
spectives of  various  other  simple  objects,  from  their  projections,  as 
in  this  example. 


lJiiOJECTIONS   AND   PERSPECTIVES    QF   CIRCULAR   BODIES.  43 


CHAPTER  VI. 

PROJECTIONS  AND  PERSPECTIVES  OF  CIRCLES,  AND  OF   BODIES   HAYING 
PARTLY  OR  WHOLLY  CIRCULAR  BOUNDARIES. 

75.  The  outlines  of  almost  all  artificial  objects  will  be  found,  by 
analyzing  them,  to  consist  of  straight  lines  and  circular  lines. 
Having  now  shown  how  to  find  the  perspectives  of  points,  straight 
lines,  and  plane  sided  figures,  both  pictorially  and  by  actual  con- 
struction, we  next  proceed  to  explain  the  construction  of  the  per- 
spectives of  circles,  and  of  various  bodies  bounded  in  part,  at  least, 
by  circles. 

EXAMPLE  5. — To  find  the  Perspective  of  a  Circle  lying  in 
the  horizontal  plane. 

The  method  by  two  planes,  with  the  vertical  or  perspective  plane 
translated  forward  before  being  revolved  back  into  the  horizontal 
plane  (71)  is  here  employed.  See  Fig.  23. 

Let  acde  be  the  horizontal  projection  of  the  given  circle.  As 
this  circle  lies  in  the  horizontal  plane,  its  vertical  projection,  a'd', 
must  lie  in  the  ground  line  LL  (45).  Now  let  the  perspective 
plane,  which  is  perpendicular  to  the  paper  at  LL,  be  translated 
forward  to  the  parallel  position  L'L',  and  then,  as  usual,  revolved 
backwards  into  the  horizontal  plane,  or  plane  of  the  paper.  Then 
take  EE'  as  the  point  of  sight,  and  let  all  the  vertical  projections 
be  shown  on  the  translated  position  of  the  perspective  plane. 
Accordingly,  a"d"  will  be  the  new  vertical  projection  of  the  given 
circle.  5E  is  the  horizontal,  and/^E'  the  vertical  projection  of  the 
visual  ray  from  the  point  £,/"  in  the  circle.  The  point  n,  where 
the  horizontal  projection  &E  meets  the  ground  line,  LL,  is  the 
horizontal  projection  of  that  point  of  the  ray  itself  in  which  it 
pierces  the  perspective  plane  (58).  The  latter  point  is  at  once  in 
the  perpendicular,  riB,  to  the  ground  line,  and  in  the  vertical  pro- 
jection /"'E'  of  the  same  ray.  Hence  the  desired  point  is  B, 
which  is  the  perspective  of  5,  f. 

[This  being  a  new  form  of  example,  the  construction  of  the  per- 
spective of  one  point  is  explained  as  minutely  as  if  it  had  not  been 


44 


LINEAR    PERSPECTIVE, 


fully  explained  already.     The  details  of  the  explanation  will  there 
fore  be  omitted  in  future  similar  constructions.] 


FIG.  23. 

The  ray  cE-cffE'  pierces  the  perspective  plane  at  C,  which  is 
therefore  the  perspective  of  c,c".  In  like  manner  the  perspectives 
of  any  other  points  can  be  found. 

By  inspection  of  the  vertical  projection,  a"dr> ',  it  appears  that 
the  extreme  visual  rays,  as  a"E',  as  seen  in  vertical  projection,  are 
those  which  proceed  from  the  opposite  ends  of  the  diameter  whose 
horizontal  projection  is  at.  Hence  rays  from  points,  as  g,  before 
that  diameter,  or  #,  behind  it,  find  their  vertical  projections,  g"E' 
and/"'E'  within  a"E'.  Hence  no  point  of  the  perspective  of  the 


PROJECTIONS    AND  PERSPECTIVES    OF    CIRCULAR   BODIES. 


45 


circle  can  appear  outside  of  the  ray  a"E',  and  therefore  the  per- 
spective must  be  tangent  to  e&"E',  at  A,  the  perspective  of  aa". 

The  similar  result,  at  the  perspective  of  tf,  is  not  shown,  as  it 
could  not  appear  distinctly  on  account  of  the  position  of  EE'. 

The  rays  whose  horizontal  projections  are  tangent  to  the  plan 
at  g  and  d,  include  the  other  rays  between  them.  Hence  all  points 
of  the  perspective  are  between  the  perpendiculars  mD  and  AC,  and 
the  perspective  is  tangent  to  these  perpendiculars  at  D  and  G. 

The  perspectives  of  tangents,  parallel  to  the  ground  line,  will 
be  tangents  to  the  perspective  and  parallel  to  I/I/.  Having  now 
six  tangents  with  their  points  of  contact,  besides  other  points,  the 
perspective  curve  can  be  very  accurately  sketched. 

76.  In  the  previous   perspectives   ol  plane-sided  figures,  which 
are  distinguished  by  well  denned  edges  and  corners,  however  viewed, 
it  will  be  observed  that  it  can  be  determined,  by  simple  inspection, 
which  edges  will  be  visible  from  the  point  of  sight.     But,  in  the 
case  of  objects  bounded  partly  or  wholly  by  continuous  curved 
surfaces,  the  consequent  partial  or  total  absence  of  limiting  edges 
makes  it  necessary  to  discover  the  visible  boundaries  by  more  or 
less  of  preliminary  construction.    Hence,  a  few  additional  defini- 
tions and  principles  are  introduced  here  for  use  in  the  following 
problems : 

77.  Other  planes  than  the  planes  of  projection,  go  by  the  general 
name  of  auxiliary  planes. 

Their  positions  are  indicated  by  their  intersections  with  the 
planes  of  projections,  called  their  traces. 

Each  of  these  traces  takes  its  name  from  the  plane  of  projection 
in  which  it  is  found. 

78.  The  point   where   either  trace  meets  the  ground  line  is 
where  the  plane  cuts  the  ground 

line ;  hence  both  traces  of  a  plane 
must  meet  the  ground  line  at  the 
same  point,  if  they  meet  it  at  all. 
The  traces  of  a  plane  will  meet 
the  ground  line  unless  the  plane 
is  parallel  to  that  line. 

79.  If,  as  in  Fig.  24,  a  plane  is 
vertical,  but  oblique  to  the   ver- 
tical plane  of  projection,  its  verti- 
cal trace,  VT,   will  be  perpendi-  Fig.  24. 
cular  to  the  ground  line,  G  L. 

80.  If,  as  in  Fig.  25,  a  plane  is  perpendicular  to  the  vertical 


46 


LINEAR    PERSPECTIVE. 


plane  of  projection,  its  horizontal  trace,  H  T,  will  be  perpendicular 
to  the  ground  line,  G  L. 

If  a  plane  is  perpendicular  to  both  of  the  planes  of  projection, 
both  of  its  traces  will  be  perpendicular  to  the  ground  line,  as  we 
have  seen  in  (72-74). 


FIG.  25. 

81.  Again;   when  a  plane  is  vertical,  that  is,  perpendicular  to 
the  horizontal  plane,  all  points  and  lines  in  it  ave  horizontally  pro-, 
jected  in  its  horizontal  trace ;  since  the  horizontal  projections  of 
points  and  lines  are  vertically  under  the  points  and  lines  themselves. 

Likewise,  when  a  plane  is  perpendicular  to  the  vertical  plane  of 
projection,  all  points  and  lines  in  it  find  their  vertical  projections  in 
its  vertical  trace. 

82.  Any  plane  containing  the  point  of  sight,  contains  an  indefi- 
nite number  of  visual  rays,  whose  directions  radiate  in  all  direc- 
tions, in  that  plane,  and  from  the  eye.     Hence  such  a  plane  is  called 
a  visual  plane. 

83.  A  visual  plane  being  thus  composed  of  visual  rays,  if  such 
a  plane  be  passed  through  a  line  whose  perspective  is  to  be  found, 
the  trace  of  that  visual  plane  on  the  perspective  plane  will  be  the 
perspective  of  the  given  line.     See  Ex.  1,  Rem.  £,  also  Fig.  10, 
where  the  plane  triangle  EAB  serves  to  mark  the  visual  plane  of 
indefinite  extent,  and  containing  the  line  AB.     A  BI  is  a  portion  of 
the  trace  of  this  plane  on  the  perspective  plane,  and  is,  therefore, 
the  perspective  of  AB. 

84.  For  the  reason  just  given  (82),  the  point  or  line  at  which  a 
visual  plane  is  tangent  to  a  curved  surface,  is  a  point  or  line  of  the 


PROJECTIONS  AND    PERSPECTIVES    OF  CIRCULAR  BODIES.  47 

visible  contour  of  that  surface.  The  perspective  of  this  visible 
contour  or  boundary,  is  the  boundary  of  the  required  perspec- 
tive. (32.) 

Ex.  6.— To  find  the  Perspective  of  a  Cylinder,  standing 
on  the  horizontal  plane.  A  cylinder,  seen  from  above,  as  it 
stands  on  the  horizontal  plane,  appears  only  as  a  circle.  As  seen 
looking  forward  at  it,  perpendicularly  to  the  vertical  plane,  its 
diameter  and  height  are  visible.  Hence  the  circle  afm,  Fig.  26, 
and  the  rectangle  n'o'p'r'  are  the  projections  of  a  cylinder  in  the 
given  position. 


JTL 


FIG.  26. 

This  being  established,  EA,  tangent  to  the  horizontal  projection 
of  the  cylinder,  is  the  horizontal  trace  of  a  vertical  visual  plune, 
tangent  to  the  cylinder  along  a  vertical  line  of  its  convex  surface  at 


48  LINEAB   PERSPECTIVE. 

h.  Likewise,  Ea  is  the  horizontal  trace  of  a  similar  plane,  tangent 
to  the  cylinder  along  a  vertical  line  at  a.  The  vertical  projections 
of  these  lines  are  h'k'  and  a'b',  and  they  are  the  projections  of  the 
visible  boundaries  of  the  convex  surface,  as  seen  from  EE'.  The 
tangent  planes  being  vertical,  their  horizontal  traces,  as  EA,  are 
the  horizontal  projections  of  both  of  the  visual  rays,  asEA-E'A'  and 
EA-E'&',  from  the  lower  and  upper  extremities  of  the  lines  of  con 
tact,  as  h-h'k'.  (81.) 

This  being  understood,  nothing  peculiar  remains  in  the  con 
struction  of  the  perspective,  ABFK,  of  the  cylinder.  Thus,  the 
perspective  of  the  point  aa'  of  the  lower  base,  is  found  by  drawing 
the  visual  ray,  aE-a'E',  which  pierces  the  perspective  plane  at  A, 
the  intersection  of  a'E'  and  #A,  perpendicular  to  GL  at  q.  Like 
wise,  K,  the  perspective  of  the  point  A,  kr  of  the  upper  base,  is  at 
the  intersection  of  &'E'  and  sK,  &'E'  being  the  vertical  projection 
of  the  visual  ray  from  A,  &',  and  sK  the  perpendicular  to  GL  from 
the  intersection  of  GL  with  AE,  the  horizontal  projection  of  the 
same  ray. 

The  perspective  bases  are  tangent,  as  at  A  and  B,  to  the  extreme 
visible  elements,  as  AB ;  for  the  visual  plane  containing  such  ex- 
treme element,  as  a — a'V  is  tangent  to  the  visual  cone  from  either 
base.  Therefore,  the  intersections,  as  AB,  of  the  visual  plane,  and 
AtfF,  or  BK,  of  the  visual  cone,  with  the  perspective  plane  are 
tangent  to  each  other,  as  at  A.  (See  Art.  85.) 

Remarks. — a.  Since  there  will,  even  when  great  care  is  taken, 
often  be  slight  instrumental  errors  in  the  construction  of  points,  the 
curves  in  the  perspective  can  be  more  advantageously  drawn  by 
carefully  connecting  a  few  carefully  constructed  points  by  easy 
curves,  than  by  finding  many  points  in  those  curves. 

b.  The  figures  in  this  book  being  designed  for  purposes  of  instruc- 
tion, necessarily  show  the  lines  of  construction  much  more  fully  than 
is  necessary  in  practice.     For  example,  in  finding  the  point  B,  Fig. 
26,  it  is  not  necessary  actually  to  draw  either  «E,  #'E',  or  ^B,  but 
only  to  mark  the  point  q  in  the  line  aE,  then  to  draw  little  frag- 
ments of  #'E'  and  $J3,  just  at  their  intersection  B. 

Likewise  in  Fig.  22,  all  that  is  essential  in  finding  O,  for  example, 
after  drawing  oE — o'E',  is  to  make  p^O  equal  to  Q^>,  and  in  the 
horizontal  liuep'pi'. 

c.  Another  matter   of  still  greater  practical  importance,  is  the 
order  in  -which  the  lines  of  construction  should  be  drawn.     All 
the  lines  necessary  for  finding  the  perspective  of  one  point,  should 
be    drawn,  before  proceeding   to   draw   those  by  which  a  new 


PROJECTIONS    AND   PERSPECTIVES    OF   CIRCULAR   BODIES.          49 

point  is  found.  Thus  in  Fig.  26,  draw  «E,  a'E',  and  <?A,  which  give 
the  point  A.  That  step  being  finished,  proceed  to  draw  cE — c'E' 
and  It  which  determine  t  y  &c. 

If,  on  the  other  hand,  all  the  lines  to  E',  for  example,  be  drawn 
before  drawing  any  to  E,  there  will  be  a  considerable  liability  to 
mistake  in  noting  wrong  intersections ;  which  could  not  possibly 
lappen  by  the  first  method  of  operating. 

d.  In  every  case  like  this,  where  points  of  the  vertical  projection 
are  shown  only  on  the  second  position  of  the  perspective  plane,  its 
first  position,  at  GL,  need  not  be  supposed  to  be  revolved  back  into 
the  horizontal  plane,  but  to  remain  perpendicular  to  the  paper  till 
after  translation  to  its  second  position  at  G'L'. 

85.  The  convex  surface  of  a  CONE,  as  V-ATB,  Fig.  27,  is  com- 
posed of  straight  lines,  which  meet  at  its  vertex,  V.  Hence  a  plane 
may  be  made  to  rest  on  this  convex  surface,  along  any  one  of  its 
straight  lines.  This  plane  will  be  tangent  to  the  convex  surface. 

When  this  plane  contains  the  point  of  sight  from  which  the  cone 
is  viewed,  it  will  be  a  visual  tangent  plane,  and  the  line  on  the  cone 
along  which  it  is  tangent,  will  be  a  boundary  between  the  visible 
and  invisible  portions  of  the  convex  surface  of  the  cone.  There 
will  evidently  be  two  such  tangent  visual  planes,  and  boundaries, 
for  any  cone. 


FIG.  27. 

In  Fig.  27,  let  E  be  the  point  of  sight  from  which  the  cone 
V-ATB  is  seen.  Then,  as  all  the  lines  of  the  convex  surface  meet 
at  the  vertex,  V,  the  two  tangent  visual  planes  through  E,  will  alsc 


50  LINEAR  PERSPECTIVE. 

pass  through  Y,  and  henee  both  will  contain  the  visual  ray  EV 
through  the  vertex. 

Now  let  HK  be  a  horizontal  plane  on  which  the  cone  V-ATB 
stands,  and  let  P  be  the  point  where  the  visual  ray  EV  pierces  this 
plane. 

It  will  then  be  evident  on  inspection  of  this  figure,  or  of  a  paper 
model  such  as  the  student  can  make ;  1° :  That  as  EP  is  a  line  com- 
mon to  both  of  the  tangent  planes,  P,  where  EP  meets  the  plane 
HK,  will  be  a  point  common  to  the  traces  (77)  of  bqth  of  the  tan- 
gent planes  upon  the  plane  HK.  2C  :  That  these  traces,  being  the 
intersections  of  planes  with  each  other,  will  be  straight  lines ;  and 
3°  :  That  as  each  visual  plane  is  tangent  to  the  cone  along  a  straight 
line  of  its  convex  surface,  the  trace  of  either  visual  plane  upon  HK 
will  be  tangent  to  the  base  of  the  cone,  which  lies  in  the  plane  HK. 

Therefore  to  find  T,  and  hence  TV,  a  line  of  contact  of  a  tangent 
visual  plane  with  the  cone,  draw  PT,  tangent  to  the  cone's  base, 
and  TV  will  be  the  cone's  apparent  contour  on  one  side,  as  seen 
from  E.  Likewise  draw  Ptf,  and  tV  will  be  the  opposite  visible 
boundary  of  the  cone's  convex  surface,  seen  from  E. 

We  will  now  proceed  to  show  this  pictorial  illustration  in  true 
projection,  with  the  perspective  of  the  cone. 

EXAMPLE  7. — To  find  the  Perspective  of  a  Cone,  standing 
on  the  horizontal  plane. 

1.° — Preliminary  explanation  of  the  projections.  Fig.  28. 
Three  planes  are  here  used,  the  horizontal  and  vertical  planes  of 
projection  ;  and  the  perspective  plane,  placed  at  right  angles  to 
both  of  them. 

The  cone  is  supposed  to  stand  on  the  horizontal  plane.  In  this 
position,  its  horizontal  projection  is  a  circle  V-TAB,  and  its  verti- 
cal projection  is  an  isosceles  triangle  V'A'B',  whose  base  equals  the 
diameter  of  the  base  of  the  cone. 

2°. —  Construction  of  the  apparent  contour  of  the  cone.  Let  E 
and  E'  be  the  projections  of  the  point  of  sight.  Then  VE  and  V'E' 
are  the  projections  of  the  visual  ray  through  the  cone's  vertex. 
This  ray  is  common  to  the  two  tangent  visual  planes  which  contain 
the  visible  limits  of  the  cone's  convex  surface.  Now  to  find  N, 
where  this  ray  pierces  the  horizontal  plane.  If  a  point  is  in  the 
horizontal  plane  its  vertical  projection  will  be  in  the  ground  line, 
hence,  conversely,  that  point,  as  N',  of  the  vertical  projection  of  a 
line,  which  is  in  the  ground  line,  is  the  vertical  projection  of  that 
point,  as  N,  which  is  in  the  horizontal  plane  (45).  Hence  the  ray  VE- 
V'E'  pierces  the  horizontal  plane  at  N,  and  by  (85)  NT  and  Ntf  art 


PROJECTIONS    AND   PERSPECTIVES    OF   CIRCULAR   BODIES. 


51 


the  horizontal  traces  of  tangent  visual  planes,  and  TV-T'V'  and 
tV-t '  V  are  their  elements  of  contact  with  the  given  cone.     These 


P' 


FIG.  28. 

elements,  with  the  portion  TAtf  of  the  base,  form  the  cone's  appa- 
rent contour. 

3°. — Now  to  find  the  perspective  of  this  contour,  that  is,  of  the 
cone.  The  preceding  topics  (1°  and  2°)  contain  all  that  is  peculiar  to 
this  problem.  The  construction  of  the  perspective  is  the  same  as  in 
previous  examples  where  three  planes  were  used.  The  construc- 
tion of  V",  only,  is  therefore  explained  to  assist  in  the  outset  of  the 
solution,  and  the  rest  of  the  figure  is  left  to  be  traced  out  by  the 
student.  The  visual  ray,  VE-V'E',  pierces  the  perspective  plane 
in  its  real  position,  PQP',  at  aa'.  After  translating  this  plane  to 
the  right,  to  any  convenient  distance,  as  at  PfQJP/,  it  is  revolved 
about  its  vertical  trace  QJY,  as  an  axis,  and  into  the  vertical  plane. 
Thus  aa'  proceeds  to  a^'  and  then  revolves  in  the  arc  a-La2-aifV 
to  V",  the  perspective  of  W.  (74.) 

By  constructing,  in  the  same  way,  the  perspectives  of  TT' ;  AA', 
the  point  of  the  base  \thich  is  rearest  to  the  eye ;  and  #',  the  per- 
spective figure  can  be  completed. 

Remark. — The  eye  being  here  placed  below  the  vertex,  N  and  E 
fall  on  the  same  side  of  the  cone,  less  than  half  of  which  is  therefore 


52 


LINEAR    PERSPECTIVE. 


visible.  When,  as  in  Fig.  27,  the  eye  is  above  the  vertex,  P  (coi  re- 
sponding to  N  in  Fig.  28)  and  E,  are  on  opposite  sides  of  the  vertex, 
and  evidently  more  than  half  of  the  cone  will  be  visible.  Let  the 
student  construct  the  perspective  of  a  cone  under  the  latter  condi- 
tion, as  above,  or  taking  the  vertical  plane  as  the  perspective  plane, 
as  in  Fig.  26. 

86.  Of  the  five  chief  solids  of  elementary  geometry  (68),  the 
cone  is  the  one  which  embraces  in  its  outline  the  three  primary 
geometrical  elements,  viz.  the  point,  straight  line,  and  circle. 
Hence  the  most  instructive  variety  of  operations  will  be  found  in 
constructing  perspectives  of  cones  in  varipus  positions,  such  as  the 
following. 

EXAMPLE  8. — To  find  the  Perspective  of  a  right  Cone 
with  a  circular  base,  whose  axis  is  parallel  to  the  ground 
line. 


FIG.  29. 

We  shall  employ  two  planes,  and  will  first  explain  the  projections 
of  the  cone.     Let  GL,  Fig.  29,  be  the  first,  and  G'L'  the  second 


^/YK 

'          Of   THf 

TJNIVEKSITY 

Of 

PROJECTIONS    AND    PERSPECTIVES    OF    CIRCULAR   BODIES.  53 

^^-~«-..     ,  -,    "— — ""^ 

position  of 'the  perspective  plane,  and  let  the  vertical  projection  be 
shown  on  this  second  position  only. 

When  the  axis  of  a  cone,  of  the  kind  here  given,  is  parallel  to 
the  ground  line,  it  is  parallel  to  both  planes  of  projection,  and  t&e 
projections  of  the  cone  will  be  simply  two  equal  isosceles  triangles, 
with  their  bases  perpendicular  to  the  ground  line.  Thus  YCD  is 
the  horizontal  projection  of  the  given  cone,  and  Y'A'B'  is  its  vcr 
tical  projection. 

According  to  (44)  A'  is  the  lowest,  and  B'  the  highest  point  on 
the  cone's  base.  In  looking  down  on  a  cone,  these  points  will  appear, 
one  directly  under  the  other  and  in  the  middle  of  the  width  of  tLe 
cone.  Hence  on  the  plan,  YCD,  the  point  B  is  the  horizontal  pro- 
jection of  both  A'  and  B'. 

In  like  manner  (44)  C  is  the  foremost  and  D  the  hindmost  point 
on  the  cone's  base,  and  C',  the  middle  point  of  the  height  of  that 
base,  is  the  vertical  projection  of  both  of  these  points. 

To  find  the  projections  of  any  other  point  in  the  base.  Assume 
/as  the  horizontal  projection  of  two  such  points,  since  a  vertical 
chord  will  contain  two  points,  one  over  the  other,  of  a  vertical  cir- 
cle, and  two  such  points  will  appear  in  plan  as  one  point.  The  ver- 
tical projection/',  of/  cannot  be  immediately  found,  since  the  line 
from /to/'  coincides  with  A'B'  and  hence  finds  nothing  to  inter- 
sect at  /',  which  must  therefore  be  found  in  some  other  way  ;  for 
it  is  a  law  of  all  constructions  in  drawing,  that  a  point  is  always 
found  as  the  intersection  of  two  known  lines. 

Accordingly,  revolve  the  front  semicircle,  BC,  of  the  base  to  the 
position  BC'7,  parallel  to  the  vertical  plane.  The  points  /will  then 
appear  at  f.  The  vertical  projection  of  the  semicircle  after  revo- 
lution, is  the  semicircle  B/"'A',  and  the  points/"  will  be  vertically 
projected  at  /'"  and  g'".  By  revolving  the  semicircle  back  to  its 
true  position,  remembering  that  the  axis  of  revolution  is  the  verti- 
cal diameter  B-B'A',  the  points/"/"'  and/"  g'"  will  revolve  back 
n  the  horizontal  arcs  whose  projections  are  f"f-f'"f  and  /'/- 
g'"g',  giving  g'  and  /'  as  the  desired  vertical  projections  of  tha 
points  whose  plan  is  /. 

The  projections  of  the  cone  being  now  fully  explained,  nothing 
peculiar  to  this  problem  remains  ;  the  construction  of  the  perspec- 
tive being  the  same  familiar  operation  already  often  repeated.  Thus, 
EE'  being  the  point  of  sight,  YE-Y'E'  is  the  visual  ray  through 
the  vertex  which  by  (58)  pierces  the  perspective  plane  at  w,  which 
is  therefore  the  perspective  of  W.  E/is  the  horizontal  projection 
of  both  of  the  rays  whose  vertical  projections  are  (?'E'  and/'E' 


54  LINEAR   PERSPECTIVE. 

These  rays  pierce  the  perspective  plane  at  H  and  F,  the  perspectives 
of/,#'  and^'.  Finding  other  points  of  the  base  in  like  manner, 
and  joining  them,  will  give  the  perspective  of  the  base.  The  per- 
spective of  the  convex  surface  of  the  cone  consists  merely  of  two 
lines  from  v,  tangent  to  FH<#,  the  perspective  of  the  base. 

Remarks. — a.  The  construction  of  the  perspective  of  the  base  i 
the  same  that  would  be  used  in  finding  the  perspective  of  any  verti 
cal  circle  which  should  be  also  perpendicular  to  the  vertical  plane. 
#.  The  same  construction  of/"  and  g'  would  be  required  in  th 
use  of  three  planes  of  projection.     The  perspective  of  such  a  circle 
by  the  method  of  three  planes  is  left  as  an  exercise  for  the  student. 
EXAMPLE  9. — To  find  the  Perspective  of  a  Cone,  whose 
axis  is  parallel  to  the  vertical  plane  only. 

All  that  is  peculiar  to  this,  and  the  following  example,  being  the 
construction  of  the  projections  of  the  cone,  they  only  will  be 
explained ;  leaving  the  construction  of  the  perspectives  as  an  exer- 
cise for  the  student. 

By  (52)  it  is  evident  that  VO  and  Y'C',  Fig.  30,  may  be  taken 
as  the  projections  of  the  axis  of  a  cone  having  the  given  position, 
VO  being  parallel  to  the  ground  line  GL.  The  axis  being  parallel 
to  the  vertical  plane,  the  base  of  the  cone  will  be  perpendicular  to 
the  same  plane  and  A'B',  perpendicular  to  V'C',  and  bisected  at  C', 
will  be  its  vertical  projection. 

Four  points  of  the  horizontal  projection  of  this  base  are  readily 
found.  A'  and  B',  the  highest  and  lowest  points,  are  horizontally 
projected  at  A  and  B,  on  the  line  ABV,  which  is  the  common  hori- 
zontal projection  of  the  axis,  and  of  that  diameter  of  the  base,  which 
is  parallel  to  the  vertical  plane.  C',  the  vertical  projection  of  the 
foremost  and  hindmost  points,  is  horizontally  projected  at  C  and  D ; 
by  making  OC  =  OD  =  A'C'. 

A  circle,  seen  obliquely,  appears  as  an  ellipse.  Accordingly  an 
ellipse,  or  a  smooth  oval  curve  representing  one,  may  now  be  traced 
through  the  points  A,C,B,  and  D,  by  the  aid  of  an  irregular  curve 
(9).  Four  intermediate  points  may  however  be  easily  found,  which 
if  accurately  located  and  regularly  distributed,  will  render  it  very 
easy  to  trace  the  required  ellipse  by  hand.  For  this  purpose,  we 
therefore  assume  o'  and  ri ',  equidistant  from  C'.  Each  of  these  is 
the  vertical  projection  of  two  points  of  the  base.  Revolve  this  base 
about  A'B',  till  it  becomes  parallel  to  the  vertical  plane,  and  o'  and 
n'  will  appear  at  o"  and  n".  o'o"  or  n'n" — perpendicular  to  A'B' — 
will  then  be  the  true  distance  of  the  points  at  o'  and  n'  before  and 
behind  the  diameter  A'B'.  Hence  from  o'  and  n'  draw  projecting 


PROJECTIONS    AND   PERSPECTIVES    OF   CIRCULAR   BODIES.  55 

lines  perpendicular  to  GL,  making  Os  =  Ot,  and  make  sp  =  sn  =» 
tr==to  =  o'o"  =  rin".  Then  o,  n,  r,  and  p  will  be  four  regularly 
distributed  points  through  which,  and  the  four  previously  found, 
the  elliptical  horizontal  projection  of  the  cone's  base  can  easily  be 
sketched  by  hand. 


FIG.  30. 

To  complete  the  horizontal  projection  of  the  cone,  merely  draw 
the  tangents  from  V  to  the  ellipse  just  drawn.  The  arc  pBn  be- 
tween these  tangents,  is  invisible,  being  on  the  under  side  of  the  cone. 

Having  now  the  complete  projections  of  the  cone,  and  the  point  of 
sight  EE',  the  perspective  can  be  found  as  in  the  previous  problems. 

EXAMPLE  10. — To  find  the  Perspective  of  a  Cone,  'whose 
axis  is  oblique  to  both  planes  of  projection. 

For  further  variety,  the  perspective  plane  is  here  taken  (Fig.  31) 
as  a  third  plane,  PLP',  perpendicular  to  the  two  principal  planes  of 


56 


LINEAR    PERSPECTIVE. 


projection.  It  is,  however,  recommended  to  the  student  to  solve 
this  problem  with  two  planes  only,  besides  the  auxiliary  plane 
whose  ground  line  is  gl /  also  to  solve  the  two  preceding  problems 
011  three  planes. 


FIG.  31. 

Let  GL  be  the  principal  ground  line,  and  YA  and  V'A'  the  pro- 
jections of  the  axis  of  the  cone.  This  axis  being  oblique  to  both 
planes  of  projection,  the  base  which  is  perpendicular  to  it,  is  oblique 
to  them  both,  also,  and  hence  will  appear  as  an  ellipse,  in  both  of 
the  required  projections,  so  that  neither  of  these  projections  will 
show  its  real  size. 

In  all  the  simpler  preceding  figures,  we  have  seen  that  at  least 
one  of  the  two  projections  shows  two  of  the  three  dimensions  of  a 
solid  in  their  real  size ;  so  here,  where  neither  of  the  required 
projections  possesses  this  property,  we  must  begin,  as  always,  with  a 
projection  upon  an  auxiliary  plane  so  situated  as  to  show  upon  it 
the  real  size  of  two  of  the  dimensions  of  the  cone. 


PROJECTIONS    AND   PERSPECTIVES    OP   CIRCULAR    BODIES.  57 

Accordingly  gl,  parallel  to  AV,  is  taken  as  the  ground  line  of  an 
auxiliary  vertical  plane,  parallel  to  the  cone's  axis ;  for  two  dimen- 
sions will  appear  in  full  size  on  such  a  plane. 

Now  any  number  of  different  vertical  projections  of  one  fixed 
point,  will  be  at  equal  heights  above  the  ground  lines  of  their 
respective  vertical  planes  (44),  and  the  two  projections  of  the  sam 
point  are  always  in  the  same  perpendicular  to  the  ground  line  (61) 
Hence  make  sV"  =  s'V  and  in  the  line  VV"  perpendicular  to  gl 
Then  make  AA"  perpendicular  to  gl,  and  mA"  =  m'A.',  and  V"A 
will  be  the  real  size  of  the  cone's  axis,  projected  on  the  auxiliary 
plane  parallel  to  it. 

Next  make  c"k"  perpendicular  to  ~V"A"  and  make  AV  =  A!'k" ; 
and  draw  VV  and  V&",  and  VV&"  will  be  an  auxiliary  projec- 
tion of  the  cone,  showing  the  true  size  of  its  altitude  V'A"  and 
diameter  c"k". 

From  this  auxiliary  projection,  make  the  horizontal  projection  as 
in  the  last  problem  ;  Aa  =  Af=  A"c" ;  also  br  =  re  =  td  =  th 
equal  to  h"ti",  and  projected  down  from  e"  and  h". 

Having  thus  completed  two  projections,  the  base  in  the  required 
vertical  projection  is  found  as  V*  was.  Thus  d'p'  —  h'ri  =  h"n  ; 
a'o  =f'q  =  A"m,  <fcc.  Then  sketching  the  ellipse  a'tie'c',  and 
drawing  the  tangents  to  it  from  V',  the  required  projections  of  the 
cone  will  be  complete. 

The  student  can  now  proceed,  as  in  previous  problems  where 
three  principal  planes  at  right  angles  to  each  other  are  used,  to 
find  the  perspective  of  this  cone  AV — A'V  on  a  perspective  plane, 
as  PLP',  at  right  angles  to  both  of  the  principal  planes  of  projec- 
tion. 

Remark. — The  operations  of  projection  applied  to  the  bases  of 
the  cones  in  the  several  preceding  problems,  are  the  same  that 
would  be  necessary  in  finding  the  perspectives  of  isolated  circles 
in  similar  positions.  Hence  separate  problems  upon  circles  hav° 
not  been  given. 

87.  Here  leaving  Cylinders  and  Cones,  whose  surfaces  are  called 
single  curved  surfaces,  because  straight  lines  can  be  drawn  in  cer- 
tain directions  upon  them,  we  pass  on  to  bodies  having  surfaces 
called  double  curved,  such  as  a  Sphere,  on  which  no  straight  line 
can  be  drawn.* 


*  As  most  double  curved  surfaces  in  the  arts  are  found  on  small  objects,  urns 
vases,  &c.,  which  may  be  drawn  by  the  eye  after  their  larger  supporting  or  surround 


58  LINEAR   PERSPECTIVE. 

EXAMPLE  11. — To  find  the  Perspective  of  a  Sphere. 

88.  This  example  is  first  taken,  since  a  sphere  is  the  simplest  poa 
sible  double  curved  surface,  inasmuch  as  the  section  of  it  made  by 
any  plane,  is  a  circle. 

From  the  fact  just  stated,  it  might  at  first  be  supposed  that  the 
perspective  of  a  sphere  would  always  be  a  circle  ;  but  not  so,  though 
it  would  always  appear  (39)  as  a  circle,  from  the  given  point  of 
sight.  For  the  visual  rays  from  the  apparent  contour  of  a  sphere 
will  always  form  a  cone,  with  a  circular  base  which  is  this  same  con- 
tour. But  the  intersection  of  this  cone  with  the  perspective  plane, 
which  will  be  the  perspective  of  the  sphere,  will  not  be  a  circle 
unless  the  perspective  plane  is  perpendicular  to  the  axis  of  this  cone, 
that  is  to  the  visual  ray  from  the  centre  of  the  sphere.  This 
statement  touches  on  the  subject  of  conic  sections,  but  the  student 
can  easily  satisfy  himself  of  its  truth  by  placing  a  paper  cone 
around  a  ball,  observing  its  circle  of  contact  with  the  sphere,  sup- 
posing its  vertex  to  be  the  place  of  the  eye,  and  then  intersecting 
the  cone  between  its  vertex  and  the  sphere  by  planes  in  various 
positions. 

There  are  two  quite  different  methods  of  determining  the  appa- 
rent contour,  whose  perspective  constitutes  the  perspective  of  the 
sphere.  The  determination  of  this  apparent  contour  forms  the  chief 
portion  of  the  solution  of  the  problem,  and  to  it  we  therefore  first 
particularly  attend. 

For  further  and  instructive  variety,  we  will  represent  one  of  these 
methods  of  finding  the  apparent  contour  on  two  planes ;  and  the 
other  on  three  planes. 

First  Method* — The  projections  of  a  sphere  will  evidently  be 
two  equal  circles,  whose  centres  will  be  in  the  same  perpendicular 
to  the  ground  line.  Then,  in  Fig.  32,  let  the  circle  with  O  for  its 
centre  be  the  horizontal  projection  of  a  sphere,  whose  centre  is  at  a 
distance  behind  the  perspective  plane  equal  to  the  distance  of  O 
>om  the  ground  line  LL.  The  equal  circle  whose  centre  is  O',  i 
he  vertical  projection  of  the  same  sphere,  and  the  height  of  O 
above  LL  is  the  height  of  the  centre  of  the  sphere  above  the  hori- 
zontal plane.  E  and  E'  are  the  projections  of  the  point  of  sight, 
and  the  vertical  plane  is  taken  also  as  the  perspective  plane. 

ing  objects  shall  have  been  found,  the  two  following  problems  may  be  omitted  at  the 
discretion  of  the  teacher. 

*  This  method  being  chiefly  valuable  as  an  intellectual  exercise  in  the  conception 
of  positions  and  motions  in  space,  it  may  be  omitted  at  the  discretion  of  the  teacher 


PROJECTIONS    AND   PERSPECTIVES    OF    CIRCULAR   BODIES. 


59 


It  is  now  evident  that,  if  a  vertical  visual  plane  EO#  be  drawn 
through  the  centre  of  the  sphere,  it  will  cut  a  vertical  great  circle 


I/ 


»$"        7" 

FIG.  32. 

from  the  sphere,  to  which  two  tangent  visual  rays  may  be  drawr. 
The  points  of  tangency  of  these  visual  rays  will,  as  such,  be  points 
of  the  apparent  contour  of  the  sphere.  But  to  show  these  rays,  the 
plane  EO£  must  be  revolved  to  a  position  parallel  to  one  of  the 
planes  of  projection.  Let  it  be  revolved  about  the  horizontal 
diameter,  ab,  of  the  sphere,  till  it  becomes  parallel  to  the  horizontal 
plane  of  projection.  The  vertical  great  circle  will  evidently  then 
appear  in  the  great  circle  hc"b,  on  ab  as  a  diameter.  The  eye  is  at 
the  vertical  distance  E'n  below  the  level,  O'w,  of  the  centre  of  the 
sphere  ;  hence,  if  the  highest  point  of  the  vertical  circle  revolve  to 
the  right,  as  shown  by  the  arrow,  EE'  will  be  found  after  revolu 


60  LINEAR    PERSPECTIVE. 

tion  at  e*,  at  the  left  of  the  axis  of  revolution  £aE,  and  at  a  pei 
pendicular  distance  from  it  equal  to  E'^. 

This  done,  e"c"  and  e"d"  are  the  revolved  positions  of  the  desired 
tangent  visual  rays,  and  their  points  of  tangency,  c"  and  d" ,  with 
the  circle  d"c"b^  the  revolved  positions  of  two  points  of  the  appa- 
rent contour.  By  revolving  the  plane  containing  c"  and  d"  back  to 
ts  vertical  position  Ea5,  c"  and  d"  will  revolve  back  about  db  as  an 
xis  in  arcs  whose  horizontal  projections  are  c"c  and  d"d,  perpendi 
cular  to  ah,  and  will  give  c  and  d  as  the  horizontal  projections  of 
these  two  points  of  the  apparent  contour. 

To  find  the  vertical  projection  of  c,  for  example.  Consider,  first, 
that  it  must  be  in  a  perpendicular  to  LL,  through  c,  and  that  it  is 
at  a  height  equal  to  cc"  above  the  level  O'n  of  the  centre  of  the 
sphere.  Hence  on  c-c'  make  c'k  =  c"c,  and  c'  will  be  the  vertical 
projection  of  the  point  of  apparent  contour  whose  horizontal  pro- 
jection is  c.  The  vertical  projection  of  d,  not  shown,  in  order  to 
simplify  the  diagram,  will  be  below  O'n  at  a  distance  equal  to  d"d. 

To  find  any  other  points  of  apparent  contour.  Intersect  the 
sphere  by  any  other  vertical  visual  plane  as  E^1,  which  cuts  from  it 
the  small  circle  whose  revolved  position,  about  pq  as  an  axis,  is 
f'g"q.  In  this  revolution,  EE'  will  appear  at  e'" ;  Eg'"  being  equal 
to  ~Ei'n  and  perpendicular  to  E<?.  The  revolved  visual  rays  d"f"  and 
e'"g"  contained  in  this  plane,  give  the  points  of  contour  f  and  g" , 
whose  true  positions,  found  as  before,  are  f  and  g.  The  vertical 
projection  of  g,g",  found  also  as  before,  is  g'.  Any  other  points  of 
contour  may  be  similarly  found. 

Finally  hh'^  the  point  of  contact  of  a  tangent  vertical  visual  plane, 
is  a  point  of  apparent  contour.  A  similar  point  may  be  likewise 
found  near  c".  Also  two  tangents  from  E',  as  EW,  to  the  vertical 
projection  of  the  sphere  will  be  the  vertical  traces  of  visual  planes 
perpendicular  to  the  vertical  plane,  as  EA  is  to  the  horizontal  plane. 
Hence  their  points  of  contact,  as  m'm,  will  be  real  points  of  apparen* 
•ontour  on  the  vertical  great  circle  through  OO'  and  parallel  to  th 
vertical  plane  of  projection. 

[This  comparatively  tedious  construction,  which  shows  how 
greatly  geometrical  problems  increase  in  complexity  as  we  leav^e 
simple  plane  sided  solids,  shows  therefore  why  elementary  works 
on  perspective  so  often  confine  themselves  wholly  to  such  solids,  and 
to  plane  figures.] 

After  finding  the  points  of  apparent  contour,  the  construction  of 
their  perspectives  is  the  work  of  a  moment.  Thus,  to  find  the  per- 
spective of  cc:.  EC  -E'c'  is  the  visual  ray  from  this  point,  and  by 


PROJECTIONS   AND   PERSPECTIVES    OF   CIRCULAR   BODIES.  61 

(58)  this  ray  pierces  the  perspective  plane  at  C,  which  is  therefore 
the  perspective  of  ccr.  By  finding  the  perspectives  of  the  other 
points  of  apparent  contour  similarly,  and  joining  them,  we  shall 
have  the  perspective  of  the  sphere  as  seen  from  EE'.  To  remove 
the  perspective  from  the  projections ;  as  before,  translate  the  per- 
spective plane  forward  to  L'L',  and  e'  an  equal  distance,  and  pro 
ceed  as  in  (Ex.  3)  to  find  the  perspective  of  cc'  at  C'  instead  of  at  C 
Remarks. — a.  The  method  just  explained  is,  from  its  nature, 
called  the  method  by  secant  visual  planes. 

b.  The  student,  as  soon  as  he  clearly  conceives  of  the  positions 
and  motions  explained  in  the  preceding  solution,  and  represented  in 
Fig.  32,  will  be  able  to  see  that  the  vertical  visual  planes,  as  Ea5, 
might  have  been  revolved  in  two  other  ways.     First:  around  a 
vertical  line  at  E  till  they  should  be  parallel  to  the  vertical  plane 
of  projection.     Then  the  vertical  circle  ab  would  appear  vertically 
projected  in  a  circle,  to  which  tangent  visual  rays  could  be  drawn 
from  E'.     Second:  These  planes  might  have  been  revolved  to  a 
similar  position,  each,  about  the  vertical  diameter,  as  at  O,  of  the 
circle  contained  in  it.     In  this  case  EE'  would  revolve  to  a  new 
position. 

c.  Again :  instead  of  vertical  visual  planes,  visual  planes  might 
have  been  passed  perpendicular  to  the  vertical  plane  of  projection. 
Any  line  through  E'  and  the  vertical  projection  of  the  sphere  would 
be  the  vertical  trace  of  such  a  plane,  and  it  might  be  revolved  in 
three  ways,  analogous  to  the  three  ways  of  revolving  Ea£,  in  order 
to  show  the  circle  and  tangent  visual  rays  contained  in  it. 

The  completion  of  the  constructions  thus  suggested,  is  left  as  a 
valuable  exercise  for  the  student. 

Second  Method. — In  this  method  we  make  use  of  these  principles. 
1  °  :  That  when  a  plane  is  tangent  to  the  surface  of  a  cone,  it  is  tan- 
gent all  along  a  straight  line  or  element  of  that  surface,  from  its 
vertex  to  its  base.  2° :  That  such  a  plane  therefore  contains  the 
vertex.  3°:  That,  therefore,  if  such  a  plane  also  contains  a  point 
in  space,  it  will  contain  the  straight  line  joining  that  point  with  the 
cone's  vertex.  4°  :  That  the  line  cut  from  the  tangent  plane  by  the 
plane  of  the  cone's  base  is  tangent  to  that  base  at  the  foot  of  the 
element  of  tangency  (85,3°).  5°:  That  a  cone  may  be  circum- 
scribed tangentially  around  a  sphere  and  will  then  have  a  circle  of 
contact  with  the  sphere.  6°,  and  lastly  :  That  the  point  where  the 
element  of  tangency  of  a  plane  and  cone  intersects  the  circle  of  tan- 
gency of  the  same  cone  with  a  sphere,  will  be  the  point  of  contact 
of  that  plane  with  the  sphere. 


62 


LINEAR    PERSPECTIVE. 


If  now  the  given  point  in  space  (3°)  be  the  point  of  sight,  the  tan 
gent  plane  will  also  be  a  visual  plane,  and  its  point  of  contact  will 
be  a  point  of  the  apparent  contour  of  the  sphere,  whose  perspective 
will  be  the  perspective  of  the  sphere. 


PJ 


FIG.  33. 

Now  in  Fig.  33,  let  Vra-A'B'd'  be  the  given  sphere,  PQF  the 
perspective  plane,  and  EE'  the  point  of  sight.  Assume  the  circle 
A;B'— AwBtf  as  the  circle  of  contact  of  an  auxiliary  tangent  cone 
(5°).  The  tangents  V'A'  and  V'B'  complete  the  vertical  projection 
of  this  cone.  EV-E'V,  the  visual  ray  through  VV,  the  cone's 
vertex,  pierces  A'R',  the  plane  of  its  base,  at  R'R-R'  being  found 
first  and  then  projected  horizontally  at  R  (3°.)  Hence  by  (4°)  Rt 
and  ~Ru  are  the  traces  of  the  tangent  planes,  to  the  opposite  sides 
of  the  cone,  upon  the  plane  of  the  cone's  base.  Then  by  (4°)  and 
(6°)  tt'  and  uu'  are  points  of  contact  of  two  visual  planes  with  the 
sphere,  and  are  therefore  points  of  the  apparent  contour  of  the 
sphere. 

This  being  determined,  we  find  the  perspectives  of  tt'  and  uu  as 


PROJECTIONS   AND   PERSPECTIVES    OF   CIRCULAR   BODIES.  63 

in  previous  cases  where  three  planes  have  been  used.  Thus,  the 
visual  ray  £E-£'E'  pierces  the  perspective  plane  at  bb'  /  which  ig 
translated  parallel  to  GL,  with  the  perspective  plane,  to  the  new 
position  PjP/  then  revolved  about  QiP/  as  an  axis,  into  the  verti 
cal  plane  at  T,  which  is  the  perspective  of  tt'.  The  perspectives  of 
>ther  points  of  the  apparent  contour  may  be  similarly  found. 

As  the  eye  is  placed,  in  this  problem,  in  the  horizontal  plane 
'E',  through  the  centre  of  the  sphere,  the  apparent  contour  of  tli 
sphere  is  evidently  a  vertical  circle,  hence  nm,  a  straight  lino 
through  t  and  u,  and  which  will  be  perpendicular  to  YE,  is  its  hori- 
zontal projection,  n  and  w,  being  on  the  horizontal  great  circle  ot 
the  sphere,  are  vertically  projected  at  ri  and  m'.  Two  points  are 
horizontally  projected  at  c,  at  t  and  at  u.  Those  at  c  are  on  the 
vertical  great  circle  A'B'<#',  and  are  vertically  projected  at  cr  and  d' . 
r'  and  s'  are  in  the  line's  t-t'  and  u-u'  and  as  far  below  the  line 
mW,  as  t'  and  u'  are  above  them. 

Remarks. — a.  This  second  method  is,  from  its  nature,  called  the 
method  of  tangent  visual  planes. 

b.  In  order  to  familiarize  the  learner  more  effectually  with  this 
beautiful  method  of  tangent  visual  planes,  located  by  the  use  of 
auxiliary  tangent  cones,  we  will  now  apply  it  to  an  object  of 
another  and  very  different  form,  and  with  the  use  of  three  planes 
of  projection. 

Among  the  comparatively  few  large  double  curved  surfaces  occur- 
ring in  the  mechanic  arts,  whose  perspectives  need  to  be  accurately 
constructed,  are  Domes,  and  concave  Spires,  &c.,  whose  perspectives 
may  be  found  as  follows. 

EXAMPLE  12. — To  find  the  Perspective  of  a  concave 
Cupola-Roof. 

Let  the  figures  with  centre  A  and  vertex  A',  Fig.  34,  be  the  projec- 
tions of  the  cupola  roof;  PQP'  the  original,  or  real  position  of  the 
perspective  plane,  and  EE'  the  point  of  sight.  The  construction  of  A' 
is,  after  previous  similar  constructions,  sufficiently  indicated  in  the 
figure.  Then  assume  BtfT-B'C'  as  the  circle  of  contact  of  an  auxiliary 
cone,  tangent  to  the  inside  of  the  cupola.  Drawing  CV  tangent 
to  the  cupola  at  C',  we  find  vr  the  vertex  of  this  cone.  Then  AE- 
v'E',  the  visual  ray  from  this  vertex,  pierces  the  plane  of  the  cone's 
base  at  R'R  (Ex.  11),  hence  Rtf  and  RT  are  the  traces,  on  this 
plane,  of  two  planes  which  are  tangent  to  this  cone  on  elements  At 
and  AT  (not  drawn)  and  hence  to  the  cupola  at  the  points  tt'  and 
TT'  (Ex.11  ;  6°).  The  perspectives  of  these  points  are  t"  and  T", 
found  by  making  c*T*  =Qc,  &c.  (Ex.  6,  Rem.  t>.) 


64 


LINEAR    PERSPECTIVE. 


At  the  base  of  the  cupola  is  a  round  edged  band,  three  points  of 
\vhich,  mm'  /  nn'  and  r'  it  is  sufficient  to  find  in  perspective,  as  at 


p' 


A 


,/'  m 

^kJ™--*/_-J 

CLE±> 


FIG.  34. 

m*,  nffy  and  r".     Through  these  points  the  perspective  of  the  cupola 
can  be  sketched. 


PERSPECTIVES    OF    SHADOWS.  65 


CHAPTER  VII. 

PERSPECTIVES    OF   SHADOWS. 

General  Principles  and  Illustrations. 

89.  Perspectives  of  shadows,  like  those  of  objects,  are  readily 
found  from  their  projections,  by  the  method  of  visual  rays  already 
explained. 

But  shadows,  being  obviously  not  independent  of  the  bodies  cast- 
ing them,  require  a  little  separate  preliminary  study,  to  show  how 
they  are  found  when  those  bodies  are  given. 

90.  The  shadow  of  a  body  on  any  surface,  is  that  portion  of  that 
surface  from  which  light  is  excluded  by  the  body. 

A  shadow  is  known  when  its  bounding  edge,  called  the  line  of 
shadow,  is  known. 

91.  Rays  of  light  from  a  very  distant  source,  as  the  sun,  fall 
upon  any  terrestrial  object  in  parallel  straight  lines. 

92.  Any  ray  which  is  intercepted  by  the  given  body  will,  evi- 
dently, if  produced  through  the  body,  pierce  the  shadow  within  its 
boundary.     Any  ray,  not  intercepted  by  the  body,  will  evidently 
pierce  the  supposed  surface  containing  the  shadow,  beyond  the  edge 
of  the  shadow.     Hence  the  line  of  shadow  is  the  shadow  of  that  line 
on  the  given  body,  at  all  points  of  which  the  rays  are  tangent  to  the 
oody. 

This  line  of  contact  of  rays,  separates  the  illuminated  from  the 
miilluminated  portion  of  the  given  body,  and  is  called  the  line  of 
shade. 

93.  Since  the  line  of  shadow  is  thus  the  shadow  of  the  line  of 
shade,  the  latter  must  always  be  found  first. 

The  line  of  shade,  from  which  shadows  are  determined,  is  found 
in  the  same  general  manner  as  the  line  of  apparent  contour,  from 
which  perspectives  are  determined  ;  viz.  by  inspection  on  most 
plane  sided  bodies,  and  by  the  aid  of  tangent  rays  of  light,  or  tan- 
gent planes  of  rays  of  light,  on  curved  surfaces. 

5 


LINEAR    PERSPECTIVE. 


94.  Practically,  shadows  are  found,  a  point  at  a  time,  and  any  one 
point  in  a  line  of  shadow  is  where  a  ray  of  light,  from  some  point  in 
the  line  of  shade  of  a  given  body,  pierces  the  surface  receiving  the 
shadow. 

Hence  it  is  obvious  that  the  form  of  a  shadow  will  depend  both 
on  the  form  of  the  body  casting  it  and  that  of  the  surface  receiving 
it,  and  also  on  the  direction  of  the  light ;  while  the  method  of  find 
ing  it  will  depend  only  on  the  form  of  the  surface  receiving  it. 

95.  Finally:   To  find  a  shadow,  we  must  have  given,  by  their 
projections,  1st.  The  body  casting  it ;  2nd.  The  surface  receiving  it ; 
3rd.  The  direction  of  the  light.     These  given,  we  may  then  con- 
struct, 1st.  The  line  of  shade  on  the  given  body ;  2nd.  The  shadow 
determined  by  that  line  of  shade.     This  done,  we  can  at  last  con- 
struct the  perspective  of  the  shadow. 

Problems  of  perspectives  of  shadows  being  thus  obviously  some- 
what tedious  and  complex,  only  a  few  simple  and  generally  useful 
ones  are  here  inserted,  as  an  introduction  to  the  subject. 

The  shadows  which  most  frequently  occur  in  perspective  drawings 
such  as  are  made  largely  for  pictorial  effect,  are  the  shadows  cast 
by  lines  in  various  positions,  on  the  ground,  and  on  the  walls  and 
roofs  of  buildings.  The  following  principles  and  examples,  therefore, 
give  elementary  illustrations  of  the  operations  necessary  in  finding 
such  shadows. 

96.  Let  AB,  Fig.  35,  be  a  slender  vertical  rod  or  wire,  and  let 
LR  represent  a  ray  of  light  drawn  through  its  upper  extremity,  B, 

and  piercing  the  horizontal 
plane  GH  at  R.  Then  R 
will  be  the  shadow  of  the 
point  B.  But  the  point,  as 
A,  in  which  a  line  meets  a 
surface,  is  a  point  of  the  sha- 
dow of  that  line  on  that  sur- 
face. Hence  AR  is  the  sha- 
dow of  AB  on  the  horizontal 
plane  GH.  But  BA  being 
vertical,  AR  is  also  the  hori- 
zontal projection  of  the  ray 
BR.  Hence,  the  shadow  of 

a  vertical  line  on  the  horizontal  plane  is  in  the  direction  of  the 

horizontal  projection  of  the  light. 

97.  By  operating  in  a  precisely  similar  manner  upon  a  line  per- 
pendicular to  the  vertical  plane  GV,  it  will  be  found  that  the  sha- 


FIG.  35. 


PERSPECTIVES    OF    SHADOWS.  67 

dow  of  such  a  line  upon  the  vertical  plane,  will  be  in  the  direction 
of  the  vertical  projection  of  the  rays  of  light. 

98.  Since  the  rays  of  light  are  parallel,  it  is  clear  that  the  shadow 
of  a  vertical  line  on  the  vertical  plane,  will,  itself,  be  a  vertical 
line  ;  likewise,  the  shadow  of  a  horizontal  line  on  a  horizontal  plane, 
will  be  parallel  to  the  line,  and,  generally,  for  the  same  reason,  if  a 
line   be  parallel  to  any  plane,  its  shadow  on  that  plane   will  be 
parallel  to  the  line  itself.     Hence,  also,  the  shadows  of  parallel  lines 
on  the  same  plane,  will  be  parallel  to  each  other. 

99.  If  the  same  line  casts  a  shadow  on  both  planes  of  projection, 
the  shadows  on  the  two  planes  must  meet  the  ground  line  at  the 
same  point.     Thus  in  Fig.  36,  let  AC  be  a  vertical  line,  long  enough 
to  cast  its  shadow  partly  on  each  plane  of  projection.     Then  R, 
where  the  shadow  AR  leaves  the  horizontal  plane  GH,  must  be  the 

beginning  of  the  shadow  RT  on 
the  vertical  plane.  For  all  the 
rays  CT,  BR,  &c.,  from  points  on 
AC,  being  parallel,  form  a  plane 
called  a  plane  of  rays,  of  which 
AR  and  RT  are  the  traces,  and  it 
has  already  been  shown  (78),  that 
the  two  traces  of  the  same  plane 
must  meet  the  ground  line  at  the 
same  point. 

FIG.  36.  Finally,  in  general,  if  the  sha- 

dow of  any  line,  as  the  top  edge  of 

a  roof,  falls  on  both  of  any  two  intersecting  surfaces,  as  the  front 
and  side  of  another  building,  these  two  shadows  will  meet  at  a 
common  point  on  the  edge  dividing  those  surfaces.  Hence,  when 
we  have  the  complete  shadow  on  one  such  surface,  this  common 
point  gives  us  one  point  of  the  shadow  on  the  other  surface. 

EXAMPLE  13. — To  find  the  Perspective  of  the  Shadow  of 
a  Square  Abacus  upon  a  Square  Pillar. 

The  method  of  two  planes  is  here  employed,  GL,  Fig.  37,  being 
the  first,  and  G'L'  the  second  position  of  the  ground  line.  The 
construction  of  the  perspective  of  the  pillar  and  its  cap  (abacus)  is 
not  shown,  it  being  exactly  like  many  previous  constructions. 
Also,  no  more  of  the  vertical  projection  of  the  object  is  made  than 
is  necessary  in  finding  its  perspective,  and  shadows. 

Rays  of  light,  like  other  lines,  being  indicated,  in  position,  by 
their  projections,  let  Al  and  A'L'  be  the  projections  of  the  ray 
through  the  front  right  hand  upper  corner,  AA',  of  the  abacus 


68 


LINEAR   PERSPECTIVE. 


This  ray  pierces  the  horizontal  plane  at  /,  whoso  vertical  projection 
which  must  be  in  the  ground  line  (45),  is  L'.     Therefore  the  point 


FIG.  37. 

•tself  is  at  £,  which  is  therefore  the  shadow  of  AA'  on  the  horizontal 
lane. 

The  shadow  of  the  lower  front  edge,  aA-a'b',  of  the  abacus, 
upon  the  front  of  the  pillar,  will  be  parallel  to  itself  (98).  When 
the  direction  of  a  line  is  known,  one  point  in  it  is  sufficient  to 
determine  it ;  hence,  to  find  this  shadow,  it  is  only  necessary  to 
pass  a  ray,  as  cb-c'd',  through  any  point,  as  cc',  of  the  lower  front 
edge  of  the  abacus,  and  to  find  where  this  ray  pierces  the  front  face 
of  the  pillar,  as  at  b,d'.  In  this  case,  by  drawing  the  ray  through 
£,  in  plan,  the  point  of  shadow,  b,d,  is  made  to  fall  on  the  right 
hand  vertical  edge,  b-b'f,  of  the  pillar.  A  line  through  d',  and 


PERSPECTIVES    OF    SHADOWS.  69 

parallel  to  a'b',  will  be  the  vertical  projection  of  the  shadow  of  #A- 
a'b'.  Drawing  the  visual  ray  J'E',  its  intersection,  D.  with  DF,  the 
perspective  of  #'/"',  will  be  the  perspective  of  d! ;  and  as  the  shadow 
is  parallel  to  the  perspective  plane,  its  perspective  will  be  parallel 
to  itself  (69,  a),  that  is  a  horizontal  line  through  D. 

Next,  drawing  the  ray  ah-a'h\  we  find  M',  the  shadow  of  ad 
the  lower,  front,  left  hand  corner  of  the  abacus,  on  the  side  surface. 
gn,  of  the  pillar,  whose  vertical  projection  is  a  line  through  hr  equal 
and  parallel  to  b'f.  The  visual  ray,  AE-A'E',  from  hh'  gives  its 
perspective,  e.  Then  eo  is  the  shadow  of  a  small  portion  of  aA- 
a'b'  upon  the  left  side  of  the  pillar. 

Now  for  the  shadow  on  the  horizontal  plane.  The  shadow  of  the 
point  AA'  is  £,  where  the  ray  A£-A'L'  pierces  that  plane.  The  ver- 
tical projection  of  I  is  I/  (45),  and  the  visual  ray,  £E-L'E',  therefore 
gives  M  as  the  perspective  of  the  point  IL'.  In  the  same  way,  find 
the  shadows  of  the  points  A,#'  and  bb',  and  join  the  latter  shadow 
with  F.  The  shadow  of  A'— AK  will  be  parallel  to  that  line,  and 
will  begin  at  I.  Hence,  as  will  fully  appear  on  making  the  construc- 
tion, L'E'  will  also  be  the  vertical  projection  of  the  visual  ray  from 
the  shadow  of  K.  Hence  ME',  up  to  DF,  is  the  visible  portion  of 
the  perspective  of  the  shadow  of  A'-AK. 

EXAMPLE  14. — To  find  the  Perspective  of  the  Shadow  of 
any  triangular  Pyramid  upon  the  Horizontal  Plane. 

In  this  problem  we  shall  employ  the  simple  principles,  that  the 
shadow  of  the  point  where  any  number  of  lines  meet  is  the  point 
where  the  shadows  of  those  lines  meet ;  and  that  the  point  in  which 
a  line  pierces  the  horizontal  plane  is  a  point  of  its  shadow  on  that 
plane. 

The  method  by  two  planes  is  employed,  and  the  construction  of 
the  perspective  of  the  pyramid,  being  the  same  as  in  many  previous 
problems,  is  briefly  indicated  in  the  diagram,  only,  Fig.  38. 

Let  ABC  be  the  plan  of  the  base  of  the  pyramid,  and  V,  that  of 
its  vertex.  V-A'B'C'  is  the  vertical  projection  of  the  pyramid 
This  vertical  projection,  being  shown  in  full  on  the  original  positioi 
of  the  vertical,  or  perspective  plane,  only  its  points,  A'B'C*  and  V ", 
are  shown,  in  the  same  relative  position,  on  the  translated  position 
of  the  same  plane,  whose  ground  line  is  G'L'.  In  fact,  after  becom- 
ing quite  at  home  in  the  subject  of  perspective,  the  student  will  see 
that  A'B'C'-V  might  have  been  omitted  altogether ;  and,  in  gene- 
ral, that  often  only  points,  and  not  lines,  of  the  projections  of 
objects  need  be  shown,  in  order  to  find  their  perspectives. 

Having,  as  in  previous  problems,  found  v-abc,  the  perspective  of 


70 


LINEAR   PERSPECTIVE. 


the  pyramid,  draw  the  ray  of  light  VR-VR'  which  pierces  the 
horizontal  plane  at  R,  projected  back  from  R'  in  the  ground  line. 


C"         B" 


FIG.  38. 

Then  R  is  the  shadow  of  W  on  the  horizontal  plane,  and  A  and 


PERSPECTIVES    OF    SHADOWS.  71 

C  being  their  own  shadows — (96 — and  the  second  principle  above 
stated— )RA  and  RC  are  the  shadows  of  VA  and  VC.  Then, 
drawing  the  visual  ray  RE-R"E',  we  find  r  for  the  perspective  of 
RR" ;  R"  being  the  vertical  projection,  R',  in  its  second  position. 
Hence  ra  and  re  are  the  perspectives  of  the  shadows  RA  and  RC, 
which  limit  the  shadow  whose  perspective  was  required. 
.  EXAMPLE  15. — To  find  the  Perspective  of  the  Shadow  of 

Dormer  Window  upon  a  Roof. 

In  this  concluding  example  of  shadows,  found  by  primitive 
methods,  we  will,  for  further  variety,  employ  the  method  by  three 
planes.  Again,  this  example  involves  the  shadows  of  lines  in  three 
different  positions,  upon  a  slanting  surface,  and  affords  the  most 
instructive  variety  with  the  fewest  lines.  Moreover,  as  the  sha- 
dows of  lines  are  determined  by  the  shadows  of  points  in  them,  and 
as  the  shadow  of  a  point  is  the  same — and  similarly  found — whether 
the  point  be  on  a  straight  line  or  curve,  a  careful  study  of  this  and 
the  two  preceding  examples  should  enable  the  student  to  find  the 
projections  and  perspective  of  any  ordinary  shadow. 

First,  now,  in  Fig.  39,  to  find  the  projections  of  the  windows  and 
roof.  To  avoid  unnecessary  lines,  only  a  small  portion  of  one  slope 
of  a  roof  is  shown,  of  which  ABCD  is  the  plan,  CDC'"D'"  is  the 
auxiliary  elevation,  showing  the  true  size  of  the  front  of  the  dor- 
mer, and  A'B'C'D'-C"D",  found  as  in  Ex.  10,  is  the  principal 
elevation.  G  and  F  are  the  plans  of  the  vertical  edges  of  the 
dormer,  whose  true  heights  I"G"  and  J"F"  appear  in  vertical  pro- 
jection at  I'G'  and  J'F'.  The  vertical  projection,  E',  of  the  peak 
of  the  gable,  FEG,  is  found  on  the  projecting  line  ENE',  by  laying 
off  NE'  equal  to  its  height,  N"E",  above  the  horizontal  plane. 
Then  draw  E'G'  and  E'F'.  NK'  parallel  to  B'C",  is  evidently 
the  trace  on  the  roof,  of  a  vertical  plane  through  the  ridge 
EK-E'K',  which  therefore  meets  the  roof  in  this  trace  at  K', 
whose  horizontal  projection  is  K.  The  points  HH'  and  LI/  are 
imilarly  found,  and  then  joined  with  KK'. 

Next,  to  find  the  projections  of  the  shadows  on  the  roof.     Le 
FP  and  F'P'  be  the  projections  of  a  ray  of  light  to  which  all  tho 
other  rays  are  parallel.     The  shadow  of  the  vertical  edge,  F-F'J', 
will  fall  in  FS-J'S',  the  trace  on  the  roof  of  a  vertical  plane  of  rays 
(99)  through  that  edge. 

The  ray  FP-F'P'  meets  this  trace  at  P',  which  is  then  horizon- 
tally projected  at  P,  and  FP-J'  P'  is  the  shadow  of  F-J'F'.  Then 
PP'  being  the  shadow  of  FF'  (94),  and  LI/  being  its  own  shadow 
(96),  LP-LT'  is  (geometrically,  for  this  shadow  is  unreal),  the  sha- 


72  LINEAR    PERSPECTIVE. 

dow  of  FL-F'L'.   The  shadows  of  parallel  lines,  on  the  same  plane, 


FIG.  39. 


PERSPECTIVES    OF   SHADOWS.  7S 

being  parallel  (98),  KR-K'R'  the  shadow  of  EK-E'K',  is  parallel  to 
LP-L'P'  and  is  limited  at  R  by  the  ray  ER.  R'  is  then  projected 
from  R.  Finally,  by  drawing  RP-R'P',  we  have  the  shadow  of 
EF-E'F'. 

Lastly,  to  find  the  perspective  of  the  roof  and  shadows,  whose 
projections  have  just  been  completed.  Let  b"A's  be  the  original, 
and  a'd'  the  translated  position  of  the  perspective  plane ;  and  let 
|  OO'  be  the  point  of  sight.  This  construction  scarcely  needs  any 
explanation,  exactly  similar  ones  having  been  often  fully  explained 
already.  One  or  two  points  only  are  mentioned,  to  acquaint  the 
learner  with  the  abbreviations  which  are  made  in  the  construction 
of  the  figure.  To  find  e,  for  example.  Draw  the  ray  EO-E'O'  and 
from  its  intersection,  n,  with  the  perspective  plane,  draw  nn\  parallel 
to  the  ground  line ;  then  make  rie=n"A.',  and  e  will  be  the  perspective 
of  EE',  since  this  is  obviously  equivalent  to  translating  n"  to  ri", 
and  revolving  it,  as  in  the  previous  unabridged  constructions. 

Having  found  d,  the  perspective  of  DD",  in  the  same  way,  also^, 
and  <7,  the  lines  dd",fj  and  gi  can  immediately  be  drawn  perpendi- 
cular to  the  ground  line,  since  they  are  the  perspectives  of  vertical 
lines,  dd"  is  limited  at  d",  simply  by  drawing  D'O'  to  u,  and  ud\ 
parallel  to  the  ground  line,  fj  and  gi  are  limited  by  their  intersec- 
tion with  ab.  a  is  the  perspective  of  the  point  AA',  which  is  its 
own  perspective,  it  being  in  the  perspective  plane,  and  a'a=A'A. 

The  perspectives  of  the  points  of  shadow  are  found  in  the  same 
manner.  Thus,  to  find  £>,  draw  the  ray  PO-P'O'  to  qqf,  and  the 
line  of  translation  q1 pf,  and  make  p'p— A'q,  which  will  give  jt>,  the 
perspective  of  PP'.  Drawing  jp,  finding  r,  the  perspective  of 
RR',  as  p  was  just  found,  then  drawing  JOT,  and  rJc,  we  sh?ll  have 
the  complete  perspective  of  the  shadow  on  the  roo£ 


GENERAL   PRINCIPLES    AND   ILLUSTRATIONS.  76 


PART  II. 

DERIVATIVE    METHODS 


CHAPTER  I. 

GENERAL   PRINCIPLES   AND   ILLUSTRATIONS. 

100.  In  all  the  problems  of  PART  I.,  we  have  found  the  perspec- 
tive of  every  point  by  one  and  the  same  primitive  and  natural 
method,  which  consists  in  finding  where  a  visual  ray  (actually 
represented)  through  any  given  point,  pierces  the  perspective  plane. 

This  method  is  primitive,  and  peculiarly  the  natural  one,  because 
it  manifestly  embodies  the  simplest  geometrical  definition  of  the 
perspective  of  a  given  point,  viz.  that  it  is  where  a  visual  ray  from 
that  point  pierces  the  perspective  plane  (35). 

It  is  true,  that  in  the  practical  application  of  this  method,  having 
revolved  the  perspective  plane  directly  back  into  the  horizontal 
plane,  a  difficulty  arose,  as  in  Figs.  16  and  IV,  from  the  confound- 
ing together  of  the  projections  and  the  perspective  in  one  place  on 
the  paper.  This  difficulty  led,^^:  to  the  translation  forward  of 
the  perspective  plane,  till  it  could  be  revolved  back  into  the  hori- 
zontal plane  so  as  to  bring  the  perspective  below  the  projections,  as 
in  Fig.  19;  second:  to  the  use  of  three  distinct  planes,  as  in  Figs. 
22,  etc.,  where  the  difficulty  of  confusion  of  figures  was  obviated 
nost  completely.  But  these  merely  particular  graphical  methods 
f  applying  the  method  of  visual  rays,  evidently  do  "not  alter  the 
method  itself,  and  we  repeat,  that  all  the  problems  of  PART  I.  were 
solved  by  the  primitive  method  of  finding  where  visual  rays,  actu- 
ally represented,  through  given  points,  pierced  the  perspective 
plane. 

101.  All  problems  whatever,  in  perspective,  might  be  readily 
solved  in  this  simple  and  beautiful  manner ;  but  by  inspection  of 
the  perspectives  thus  found,  certain  peculiarities  may  be  discovered, 
which,  on  examination,  lead  to  other  methods,  hence  called  deriva- 


76  LINEAR   PERSPECTIVE. 

tive ;  or,  because  the  visual  rays  are  no  longer  represented,  com 
paratively,  artificial  methods. 

The  advantages  of  knowing  several  methods,  which  will  soon 
appear,  are  chiefly  two  :  1°  Abbreviation  of  the  operations  of  con- 
struction. 2*  Provision  of  checks  upon  inaccuracy. 

It  has  already  been  shown  by  experimental  proof,  in  previous 
constructions,  that  any  lines,  whether  vertical  or  horizontal,  which 
are  parallel  to  the  perspective  plane,  have  their  perspectives  parallel 
to  each  other,  and  to  the  lines  themselves.  Also  that  the  perspec- 
tives of  all  lines  which  are  perpendicular  to  the  perspective  plane, 
meet  at  the  vertical  projection  of  the  eye.  (Fig.  16.)  These  two 
results  have  also  already  been  separately  proved  to  be  true,  (Ex. 
1.  Hems.  a,b.)  but  by  now  considering  them  in  connexion  with  a  few 
others,  we  shall  arrive  at  a  body  of  principles  by  which  perspec- 
tives of  objects  can  be  found  by  the  derivative  methods,  which  it 
is  the  object  of  this  second  "  PART  "  to  explain. 

102.  In  standing  on  a  vast  plane,  such  as  a  natural  plain,  its 
remotest  visible  limit  appears  as  a  horizontal  line  on  a  level  with 
the  eye.  The  reason  of  this  is  evident  from  Fig.  40.  Let  E  be  the 


jp~~  b  <*-  H 

FIG.  40. 

place  of  the  observer's  eye,  looking  forward  in  the  direction  ER, 
parallel  to  the  ground  HP.  In  taking  successive  points  on  the 
ground,  as  «,  #,  and  P,  at  greater  and  greater  distances  from  the  ob- 
server, standing  at  H,  the  visual  rays  aE ....  PE,  &c.,  become  more 
and  more  nearly  horizontal,  and  finally,  when  a  ray  comes  from  an 
indefinitely  remote  point  on  the  ground,  its  direction  cannot  be  dis- 
tinguished from  that  of  the  horizontal  visual  ray  ER.  Hence,  as  the 
apparent  position  of  objects  depends  on  the  direction  of  the  visual 
rays  entering  the  eye  from  them,  the  very  remote  limit  of  any  level 
plane  appears  as  a  horizontal  line,  on  a  level  with  the  eye. 

103.  The  indefinitely  remote  limit  of  a  natural  plain,  or  horizon- 
tal geometrical  plane,  is  called  the  horizon  /  hence  a  line  parallel 
to  the  ground  line,  and  through  the  vertical  projection  of  the  point 
of  sight,  is  the  perspective  of  the  horizon. 


GENERAL    PRINCIPLES    AND    ILLUSTRATIONS.  77 

Such  a  line  is  called  the  horizontal  line,  or  the  horizon  of  the  pic- 
ture. 

104.  It  follows  from  this,  that  the  remotest  visible  limit  of  all 
lines  in  such  a  plane,  that  is  of  all  horizontal  lines,  will  appear  to 
be  in  the  "  horizontal "  line,  which  represents  the  remotest  limit  of 
this  plane. 

105.  Now  the  remotest  visible  limit  of  the  plane  supposed,  is 
literally  its  vanishing  line,  and,  likewise,  the  remotest  visible  point 
of  any  line  in  that  plane,  is  its  vanishing  point.     The  representation 
of  this  line,  or  point,  on  the  perspective  plane,  is  the  perspective  of 
such  line  or  point,  and  is,  according  to  the  last  two  articles,  a  line 
or  point  on  the  perspective  plane,  and  at  the  height  of  the  eye. 

106.  The  perspective  of  a  vanishing  line  or  point  being  of  con- 
stant use  in  the  construction  of  perspectives,  while  the   original 
indefinitely  distant  real  vanishing  line,  or  point,  is  not,  the  former 
is,  for  brevity,  itself  termed  the  vanishing  line,  or  point. 

Hence  we  have  these  principles :  1°.  The  vanishing  line  of  any 
horizontal  plane,  is  a  horizontal  line,  drawn  on  the  perspective  plane 
and  at  the  height  of  the  eye.  2°.  Any  horizontal  line  has  its 
vanishing  point  in  the  "horizontal"  line. 

107.  Similar  reasoning  might  be  applied,  and  with  corresponding 
conclusions,  to  vertical,  or  oblique  planes,  but  as  we  do  not  find  in 
Nature  real  planes  of  indefinite  extent,  everyway,  in  these  positions, 
it  will  be  sufficient  to  consider  the  vanishing  points  of  lines,  only,  in 
any  direction. 

108.  The  visual  ray  from  the  indefinitely  distant,  or  remotest 
visible  limit  of  an  unlimited  line,  will  evidently  appear  to  be  parallel 
to  that  line,  and  the  intersection  of  this  ray  with  the  perspective 
plane,  is  the  perspective  of  that  remote  or  real  vanishing   point. 
This  intersection  itself  (106)  is  practically  called  the  vanishing 
point,  in  making  perspective  drawings ;  and  will  be  so  called  in  the 
following  pages.     Also,  a  visual  ray  which  is  parallel  to  one  line,  is 

arallel  to  all  others,  which  are  parallel  to  that  one.  Hence  to  find 
the  vanishing  point  of  any  line  or  group  of  parallel  lines,  we  have 
the  following  rule.  Find  where  a  visual  ray,  parallel  to  the  given 
lines,  pierces  the  perspective  plane  ;  the  point  thus  found  will  be  the 
required  vanishing  point. 

109.  Illustration.  Let  PP,  Fig.  41,  represent  the  perspective  plane, 
and  L,L,L,  three  parallel  lines  in  any  direction.  These  lines  will  appa 
rently  meet,  and  so  maybe  considered  as  meeting,  at  an  indefinitely 
great  distance,  and  the  visual  ray  YE  from  their  distant  apparent 
intersection,  will,  for  any  short  distance,  as  EV,  be  sensibly  parallel 


LINEAR   PERSPECTIVE. 


to  them.  But  V,  the  intersection  of  this  ray  with  the  perspective 
plane,  is  the  perspective  of  that  intersection.  That  is,  V  is  the 
vanishing  point  of  L,L,L, 


FIG.  41. 

Observe  finally,  that  as  parallel  lines  themselves  appear  to  meet 
at  their  indefinitely  remote  point,  so  their  perspectives  will  meet  at 
their  vanishing  point  on  the  perspective  plane,  which  is  the  perspec- 
tive of  their  real  vanishing  point  in  space.  Thus,  if  aV,  5V,  and  cV 
are  the  perspectives  of  L,L,L,  they  will  meet  at  V. 

110.  In  general,  if  any  number  of  lines  meet  at  any  point,  their 
perspectives  will  evidently  meet  at  the  perspective  of  that  point. 

EXAMPLE  1. — Let  it  be  required  to  find  the  vanishing 
point  of  several  Telegraph  Wires  which  go  over  a  hill. 

In  Fig.  42  let  AA'  and  BB'  be  two  successive  poles,  carrying  two 
wires.  AB  is  the  plan  of  both  of  these  wires.  Let  CC'  and  DD' 
be  another  pair  of  poles,  of  a  line  of  single  wire,  and  let  EE'  be  the 
position  of  the  eye.  Then  EY,  parallel  to  AB  or  CD,  and  E'V, 
parallel  to  A'B'  or  C'D',  are  the  projections  of  the  visual  ray,  parallel 
to  these  wires,  and  therefore  giving  the  perspective  of  an 
indefinitely  remote  point  upon  them.  This  ray  meets  the  perspec- 
tive plane  at  V  (58),  which  is  therefore  the  vanishing  point  at 
which  the  perspectives  of  the  wires  will  meet. 

111.  From  the  general  case  just  considered,  in  illustration  of  the 
general  principle  of  (108)  let  us  proceed  to  find  the  location  of  the 
vanishing  points  of  groups  of  parallels,  having  particular  positions 
with  respect  to  the  perspective  plane. 

First.  It  follows  directly  from  the  rule  (108),  that  all  lines  which 
are  parallel  to  the  perspective  plane  have  no  vanishing  point. 
Hence  their  perspectives  will  be  parallel  to  themselves.  That  is, 
the  perspectives  of  vertical  lines,  for  example,  will  be  vertical,  as 


GENERAL   PRINCIPLES    AND   ILLUSTRATIONS.  79 

seen  in  (Fig.  16,  etc.,  PART  I.)  Also,  if  lines  are  parallel  to  the 
ground  line,  their  perspectives  will  be  parallel  to  the  ground  line, 
as  also  seen  in  Fig.  16. 


FIG.  42. 

Second.  It  also  follows  from  (108),  that  all  horizontal  lines  nave 
their  vanishing  points  in  the  horizontal  line,  or  horizon  (103). 

112.  In  particular,  among  horizontal  lines,  we  notice  those  which 
are  also  perpendicular  to  the  perspective  plane ;  and  those  which 
make  an  angle  of  45°  with  the  perspective  plane.  The  former  are 
called  perpendiculars,  and  the  latter,  diagonals. 

EXAMPLE  2. — To  find  the  vanishing  point  of  a  Perpendicu- 
lar, and  of  a  Diagonal. 

See  Fig.  43,  where  DC  is  the  ground  line,  EE'  the  point  of  sight, 
and  D'A  the  horizontal  line. 

By  (48),  when  a  line  is  perpendicular  to  the  vertical  plane,  its 
vertical  projection  is  a  point,  and  its  horizontal  projection,  a  line, 
perpendicular  to  the  ground  line.  Therefore  db  is  the  horizontal, 
and  a'  the  vertical  projection  of  a  perpendicular,  at  the  height  aa' 
above  the  horizontal  plane.  Likewise  Ee  is  the  horizontal,  and  E' 
the  vertical  projection  of  a  visual  ray,  parallel  to  db-a' .  This 
visual  ray  pierces  the  perspective,  or  vertical  plane  at  E',  which 
is  therefore  the  vanishing  point  of  db-a'  and  of  all  perpendiculars 
(108)  while  EE'  remains  as  the  place  of  the  eye. 


80 


LINEAR   PERSPECTIVE. 


113.  The  point  E',  the  vertical  projection  of  the  point  of  sight,  is 
usually  known  among  artists  as  the  centre  of  the  picture  /  since  in  a 
picture  of  equal  interest  throughout,  it  should  be  in  the  centre,  of 
the  horizontal  width,  at  least,  of  the  canvas.     Therefore  we  say  that 
the  vanishing  point  of  perpendiculars  is  at  the  centre  of  the  picture. 
E  is  often  called  the  station  point. 

114.  To  return  now  to  the  diagonal.     By  (51),  when  a  line  is 
parallel  to  the  horizontal  plane  only,  its  vertical  projection  is  paral- 
lel to  the  ground  line,  hence  (112),  making  ac  =  ab,  abc  —  acb  = 
45°,  and  be  will  be  the  horizontal  projection  of  a  diagonal  through 
the  point  5,  a'  and  a'c'  will  be  its  vertical  projection.     Then  ED- 
ET)'  is  the  parallel  visual  ray  which  pierces  the  perspective  plane 
(58),  at  D'.     Hence  D'  is  the  vanishing  point  of  bc-a'cf  and  of  all 
other  diagonals  (108). 


D' 

I 

E' 

V 

A 

1 
{ 

e              a 

\ 

D\ 


\ 


\ 


\. 


FIG.  43. 

115.  Observe  in  Fig.  43,  that  Ee  =  eD  =  E'D',  that  is,  the  dis- 
tance from  the  centre  of  the  picture  to  the  vanishing  point  of  dia- 
gonals is  equal  to  the  distance  of  the  eye  from  the  perspective  plane. 
Hence,  having  either  E  or  D'  given,  with  E',  we  can  find  the  other  of 
these  points.    Thus,  having  E'  and  D,'  make  eE  =  E'D'  which  gives 
E ;   and  having  E  and  E'  given,  make  E'D'  =  Ee,  which  gives  D'. 

116.  The  point  in  which  a  line  pierces  the  perspective  plane,  is  a 
point  of  its  perspective  ;  for  the  visual  ray  from  that  point  pierces 
the  perspective  plane  at  its  outset.    Also,  as  follows  from  (109),  the 
vanishing  point  of  a  line  is  a  point  of  its  perspective.     Moreover, 
two  points  determine  a  straight  line,  hence  the  perspective  of  a 
straight  line  is  a  line  joining  its  van  ishing  point  with  the  point 
where  it  pierces  the  perspective  plane. 


GENERAL   PRINCIPLE    AND 


81 


Thus,  in  Fig.  44,  the  perpendicular  a5-a'"pierces  the  perspective 
plane  at  a'/  and  the  diagonal,  at  c' /  hence,  if  we  draw  a'E',  it  will 
be  the  perspective  of  this  perpendicular,  and  if  we  draw  c'D',  it 
will  be  the  perspective  of  the  diagonal,  bc-a'c'. 


\ 


•:m 

FIG.  44. 

See  also  a  pictorial  illustration  in  Fig.  41,  for  lines  in  any  direc- 
tion. There  the  three  parallels  meet  the  perspective  plane  at  a,  b, 
and  c,  and  Y  being  their  vanishing  point  (109)  aV,  5Y,  and  cV  are 
their  perspectives. 

117.  It  follows  from  (110)  that  if  two  lines  intersect  at  a  point, 
their  perspectives  will  intersect  at  the  perspective  of  that  point, 
that  is,  the  intersection  of  the  perspectives  of  two  lines,  is  the  per- 
spective of  the  intersection  of  the  lines  themselves.     Hence  in  Fig. 
44,  B,  the  intersection  of  the  perspectives  of  the  perpendicular  ab- 
a'  and  diagonal  bc-a'c',  is  the  perspective  of  the  point  b,  a'  from 
which  both  of  these  lines  originated. 

Particular  Derivative  Methods. 

118.  It  is  now  apparent  that,  by  the  principles  of  (116)  and  (117) 
the  perspective  of  any  point,  and  hence  of  any  object,  can  be  found 
without  the  use  of  any  visual  rays. 

Derivative  methods,  then,  consist  in  substituting  for  the  visual 
ray  from  any  given  point,  any  two  lines  containing  that  point ;  and 
in  finding  their  perspectives,  by  joining  their  intersections  with  the 
perspective  plane,  with  their  vanishing  points  (116).  The  intersec- 
tion of  the  perspectives  of  these  lines  will  then  be  the  perspective  of 
the  given  point  (HV). 

119.  Since  all  parallel  lines  have  the  same  vanishing  point   (10S) 


82  LINEAR    PERSPECTIVE. 

it  will  obviously  abridge  the  constructions  to  use  auxiliary  lines  in 
parallel  sets.  This  being  clear,  it  further  appears,  that  no  auxiliary 
lines  are  so  universally  simple  and  convenient  as  diagonals  and  per- 
pendiculars ;  first:  because  the  centre  of  the  picture,  which  is 
always  given,  being  the  vanishing  point  of  perpendiculars,  no 
vanishing  point  need  be  constructed  for  them  ;  second,  because  the 
distance  D'E'  from  the  centre  of  the  picture  to  the  vanishing  point 
of  diagonals  is  equal  to  the  distance,  eE,  of  the  eye  from  the  per- 
spective plane,  Fig.  43  ;  so  that  if  the  latter  is  given,  the  former  is 
immediately  known,  and  if  it  is  not  given,  E'D'  can  be  assumed  at 
pleasure. 

Foremost  therefore  among  derivative  methods,  is  the  method  of 
diagonals  and  perpendiculars,  as  explained  and  illustrated  in  (112 
to  117)  all  of  which  is  based  on  (110). 

120.  The  only  other  derivative  method,  which  need  be  mentioned, 
is  one  which  is  applicable  to  bodies  bounded  by  straight  lines,  which 
are  arranged  in  parallel  groups,  as  in  a  square  prism.     In  this  case, 
the  lines  of  the  object  itself  may  be  put  in  perspective  by  (116). 
The  intersections  of  their  perspectives  will  then  be  the  perspectives 
of  the  corners  of  the  object. 

Derivative  methods,  exclusively,  are  generally  used  in  connection 
with  two  planes,  only,  of  projection. 

121.  We  will  now  close  this  chapter  with  three  fundamental  illus- 
trative examples,  showing,^?-^,  how  to  find  the  perspective  of  any 
line  whatever  by  its  vanishing  point  and  point  of  intersection  with 
the  perspective  plane ;  second,  how  to  find  the  perspective  of  any 
object  by  the  method  of  diagonals  and  perpendiculars  ;  and,  third, 
how  to  find  the  perspective  of  a  plane  sided  object  by  finding  the 
vanishing  and  intersection  points  of  its  own  edges. 

EXAMPLE  3.— To  find  the  Perspective  of  a  Straight  Line 
in  any  position,  oblique  to  both  planes  of  projection,  by 
»ts  vanishing  point  and  intersection  with  the  perspective 
lane. 

Figs.  45  and  46.  To  familiarize  the  student  more  fully  with  this 
problem,  and  so  to  render  the  conception  of  positions  in  space,  cor- 
responding to  given  projections,  more  easy,  two  different  lines  have 
been  taken  in  the  above  figures,  while  for  more  ready  comparison, 
like  points  are  lettered  with  the  same  letters  in  both  figures. 
Accordingly,  ab-a'b',  in  both  cases,  is  a  line  behind  the  perspective 
plane,  as  usual.  Its  extremity  aa'  is  at  the  distance  ac  behind  the 
perspective  plane,  and  height,  a'c  above  the  horizontal  plane,  b, 
being  in  the  ground  line,  is  the  horizontal  projection  of  that  point 


GENERAL   PRINCIPLES    AND    ILLUSTRATIONS. 


83 


of  the  given  line,  which  is  in  the  vertical  plane,  that  is,  the  point 
b'.  That  is  ab-a'b'  pierces  the  vertical,  or  perspective  plane  at  V 
which  is  therefore  one  point  of  the  perspective  of  this  line  (116). 


a 


FIG.  45. 


Again:  Ew-EV,  parallel  to  ab-a'b' ,  is  the  parallel  visual  ray 
from  the  infinitely  distant  point  of  ab-a'b'  (108).  Hence  vf,  where 
this  ray  pierces  the  perspective  plane,  is  the  perspective  of  that  infi- 
nitely remote  point.  That  is,  v'  is  the  vanishing  point  (106)  of  ab~ 
a'b'.  Hence  by  (116)  v'b'  is  the  perspective  of  ab-a'b'. 

Remarks,  a.—  As  v'  is  the  perspective  of  an  infinitely  distant 


84 


LINEAR   PERSPECTIVE. 


point  on  ab—a'b',  v'b'  is  the  perspective  ofab-a'bf  produced  to  a& 
infinite  length,  from  W,  back  from  the  perspective  plane. 

b.  As  a  further  exercise,  let  the  student  take  lines  in  othei 
positions.  Thus  in  Fig.  45  let  the  given  line  have  such  a  position 
that  a'b'  shall  be  its  horizontal,  and  ab  its  vertical  projection,  and 
then  find  its  perspective,  as  before. 

EXAMPLE  4. — To  construct  the  Perspective  of  a  Tower 
and.  Spire,  by  diagonals  and  perpendiculars. 


FIG.  47. 

Fig.  47,  let  PBE  be  the  plan  of  the  tower,  and  FGN  of  the  spire 
whose  vertex  is  A.     Let  LL  be  the  ground  line,  taken  through  the 


GENERAL   PRINCIPLES    AND   ILLUSTRATIONS.  8£ 

corner  B,  which  indicates  the  real  position  of  the  perspective  plane, 
L'L'  indicates  the  position  of  the  perspective  plane  after  translation 
forward.  CD  is  the  horizon,  and  C  the  centre  of  the  picture.  The 
vanishing  point  of  diagonals  is  assumed  on  CD,  at  the  left  of  C,  and, 
in  this  case,  beyond  the  limits  of  the  picture. 

[This  preliminary  explanation  is  substantially  common  to  most  of 
the  following  problems,  and  is  therefore  to  be  understood  though 
not  repeated.] 

Since  the  vertical  edge  at  B  is  in  the  perspective  plane,  it  is  its 
own  perspective,  hence  its  vertical  projection,  B'C',  which  shows 
the  true  height  of  the  tower,  is  also  its  perspective.  To  find  the 
perspective  of  either  of  the  other  visible  vertical  edges,  as  the  one 
at  E,  draw  the  diagonal  E5  from  the  corner,  E,  of  the  base  of  the 
tower.  E'&'  is  the  vertical  projection  of  this  diagonal,  since  it  is  in 
the  horizontal  plane ;  it  pierces  the  perspective  plane  at  5',  and  b'D 
(D  meaning  the  vanishing  point  of  diagonals,  not  shown)  is  its 
perspective.  Em,  the  perpendicular  from  the  same  point  E,  pierces 
the  perspective  plane  at  E',  and  E'C  is  its  perspective.  Hence  6, 
the  intersection  of  #'D  and  E'C,  is  the  perspective  of  E,  considered 
as  in  the  lower  base  of  the  tower,  p  is  found  from  P  in  a  precisely 
similar  manner.  Then  draw  j»B'  and  We. 

To  find  any  point,  as  A,  of  the  top  of  the  tower.  E&,  considered 
as  the  diagonal  from  the  point  E  in  the  top  of  the  tower,  pierces  the 
perspective  plane  at  the  true  height  of  the  tower,  that  is  at  c',  in 
the  horizontal  line  through  C',  since  diagonals  are  always  horizontal 
lines  (112).  Then  c'D  is  the  perspective  of  this  diagonal.  The 
perspective  of  the  perpendicular  from  the  upper  point,  E,  is  not 
needed,  since  the  perspective  of  the  vertical  edge  at  E  is  known  to 
be  a  vertical  line  through  e.  Hence  A,  the  intersection  of  eh,  drawn 
perpendicular  to  the  ground  line,  with  c'D,  is  the  perspective  of 
the  top  corner  of  the  tower  at  E.  q  is  found-in  the  same  manner 
Then  draw  qCr  and  C'A. 

To  find  any  point,  as/,  in  the  perspective  of  the  base  of  the  spire 
f  is  the  perspective  of  F,  and  the  plane  of  the  base  of  the  spire  is 
the  same  as  that  of  the  top  of  the  tower ;  hence  the  diagonal,  Frf, 
and  the  perpendicular,  F^,  pierce  the  perspective  plane  in  the  hori- 
zontal line  through  C',  at  d'  and  F',  respectively.  Then  d'D  is  the 
perspective  of  Fc?,  and  F'C,  that  of  Yg ;  and/,  the  intersection  of 
these  perspectives,  is  the  perspective  of  F.  The  top  of  the  towei 
being  above  CD,  the  level  of  the  eye,  the  base  of  the  spire  is  invi- 
sible. The  perspectives  of  G  and  N  are  found  in  the  manner  just 
described. 


80  LINEAR    PERSPECTIVE. 

Finally,  the  height  of  K'm'  above  the  ground  line,  represents  the 
height  of  the  top  of  the  spire.  Then  the  diagonal,  Am,  and  per- 
pendicular, An,  pierce  the  perspective  plane  at  m'  and  A'.  m'D  is 
the  perspective  of  Am,  and  A'C,  that  of  An.  Hence  a  is  the  per 
spective  of  A.  Now  join  a  with  f,  and  the  other  points  of  the  base 
of  the  spire,  limiting  the  lines  thus  drawn  by  qC'  and  C'A,  and  the 
required  perspective  will  be  complete. 

122.  Various  miscellaneous  points,  which  naturally  arise  in  the 
mind  of  a  beginner,  are  most  conveniently  disposed  of  hqre,  after 
the  progress  thus  far  made  with  primitive  and  derivative  methods. 
They  are  therefore  discussed  in  the  following — 

Jtemarks. — a.  The  statements  in  (101)  can  now  be  made  more 
intelligible.  First.  Derivative  methods  abridge  the  labor  of  con- 
struction: first : — through  the  partial  omission  of  the  projections^ 
as  seen  in  the  above  example,  where  the  vertical  projection  was  not 
required,  because  the  auxiliary  lines,  being  horizontal,  will  always 
pierce  the  perspective  plane  at  the  height  of  the  points  from  wrhich 
they  are  drawn,  and  these  heights  can  always  be  indicated  by  set- 
ting them  off,  as  b'c'  was,  equal  to  B'C',  the  real  known  height  of 
the  top  of,  the  tower  ;  second  : — by  the  provision  of  common  van- 
ishing points  for  all  parallel  lines,  so  that,  1°, — only  one  other 
point  in  each  indefinite  line,  need  be  found  ;  and  2°, — so  that  any 
particular  point,  as  h,  on  such  a  line,  can  be  found  by  a  single  auxi- 
liary, as  c'D. 

Second.  Derivative  methods  also  conduce  to  accuracy  /  first : — 
by  providing  against  errors  arising  from  very  acute  intersections 
in  the  lines  of  construction.  See  PART  I.,  Fig.  19,  where,  though 
the  intersection  at  A  is  well  defined,  that  at  F,  and  especially  at  the 
perspectives  of  c"  and  d"  (not  shown)  are  not.  Whenever,  there- 
fore, another  method  fails  to  give  well-defined  intersections,  that  of 
diagonals  and  perpendiculars  will  be  generally  found  available. 
Second :  by  the  provision  against  distortion  of  apparent  propor- 
tions, which  is  afforded  by  vanishing  points.  It  is  a  matter  of 
familiar  experience,  that  all  receding  parallels  in  the  same  group 
appear  to  vanish  at  the  same  point,  and  in  a  drawing,  where 
vanishing  points  are  employed,  their  perspectives  will  likewise 
vanish.  But  if  no  vanishing  points  are  used,  so  that  the  perspec- 
tive of  each  line  of  a  parallel  set  is  found  hidependently  of  the 
others,  by  finding  two  points  in  it,  it  may  happen  that  these  per- 
spectives, if  produced,  will  not  meet  at  one  point.  Errors  in  the 
true  relative  direction  of  perspectives  are  far  more  offensive  to  the 
eye  than  the  less  obvious  errors  in  the  absolute  place  of  single  points. 


GENERAL   PRINCIPLES   AND   ILLUSTRATIONS.  87 

b.  The  disuse  of  vertical  projections  which  the  method  of  diago- 
nals and  perpendiculars  allows,  is  another  advantage  of  that  method 
over  those  in  which  the  auxiliary  lines  might  not  be  horizontal. 

c.  The  question  naturally  occurs  to  a  practical  inquirer,  "  how 
shall  I  represent  an   object  of  given  dimensions,  viewed  from   a 
given  distance,  and  in  a  given  direction"     See   Fig.    47.     If  in 
practice  the  distances  from  P  and  E  to  the  observer  be  measured, 
the  exact  relative  position  of  the  tower  and  the  observer  will  be 
known,  and  so  can  be  laid   down   on  paper.     This  done,    we   can, 
from  a  given  position,  look  straight  forward  towards  the  centre  01 
an  object,  as  shown  by  the  line  CA  in  the  figure,  or  we   can   turn 
and  look  towards  the  right  or  left  of  the  centre  so  as  to  see  the 
object  partially  by  a  sidewise  glance  of  the  eye. 

The  clearest  view  is  obtained  in  the  former  case,  but  in  any  case 
the  perspective  plane  is  supposed  to  be  perpendicular  to  the  direc- 
tion of  vision.  Thus,  if  the  spectator  at  E,  Fig.  48,  observe  O, 
among  other  things,  while  looking  in  the 
direction  Ee,  the  perspective  plane  PQ 
should  be  perpendicular  to  Eey  simply 
because  this  is  its  simplest  and  most  natu- 
ral position. 

This  being  understood,  make  eD  =  eE, 
to  find  the  vanishing  point  of  diagonals 
FIG.  48.  (115)  or,  in  Fig.  47,  lay  off  from  n,  on 

AnC  produced,  a  distance  equal  to  CD,  to 

find  the  position  of  the  observer,  or  horizontal  projection  of  the 
point  of  sight,  often  called  the  station point(ll 3). 

d.  But,  further,  in  representing  large   objects    truly,  all   these 
dimensions  and  distances,  just  spoken  of,  must  be  reduced  uniformly, 
so  as  to  be  shown   at  all,  and  in  true  proportion,  on  paper.     In 
other  words  they  must  be  drawn  to  a  scale. 

For  example,  let  it  be  supposed  that  in  Fig.  47,  all  given  parts 
are  to  be  shown  on  a  uniform  scale  of  five  feet  to  an  inch,  i.e.  Jive 
feet  on  the  real  object,  to  one  inch  on  the  drawing.  On  any 
straight  line  as  XY,  lay  off  two  or  more  inches,  and  divide  each 
inch,  as  shown,  into  five  equal  parts.  Each  of  these  parts  will 
therefore  represent  one  foot,  and  hence,  in  connexion  with  the 
drawing,  may  be  called  one  foot.  Let  the  left  hand  one  of  these 
feet  be  subdivided  into  twelfths  (fourths  only  are  shown)  which 
will  be  inches.  Any  other  scale  is  made  in  a  similar  manner. 
Having  such  a  scale,  its  zero  point  is  at  the  right  hand  end  of  the 
divided  foot.  If  then,  the  tower  is  5  ft.  9  ins.  square,  as  at  BP  and 


88  LINEAR    PERSPECTIVE. 

BE,  extend  the  dividers  from  the  point  marked  five  to  the  3  inch 
point  between  6  and  12,  which  will  be  five  feet  nine  inches,  on  a 
scale  of  five  feet  to  one  inch.  So,  if  the  object  be  11  ft.  6  ins.  high, 
make  the  line  A.'m'  at  this  perpendicular  distance  from  the  ground 
line  L'L'. 

It  thus  appears  that  in  using  any  scale,  thus  constructed   and 
numbered,  no  calculations  need  be  made,  since  we  take  up  in  th 
dividers  the  same  number  of  scale  feet  and  inches,  that  there  art 
of  real  feet  and  inches  in  any  given  line  to  be  represented.     The 
question  of  Hem.  c  is  thus  fully  answered. 

e.  It  is  a  familiar  fact  that  the  apparent  size  of  an  object 
decreases  with  its  increased  distance  from  the  eye,  but  the  term 
apparent  size  is  really  a  little  ambiguous,  on  account  of  the  interfer- 
ence of  knowledge  with  sense  impressions.  Thus,  when  I  see  a 
whole  house  through  one  window  pane,  I  perceive  that  the  appa- 
rent size  of  the  house  is  less  than  that  of  the  pane,  and  it  is  so 
because  the  image  of  the  house  on  the  retina  of  the  eye,  which  is 
what  determines  its  real  apparent  size  to  simple  sense,  is  less  than 
that  of  the  pane.  But  I  know  that  the  house  is  much  larger  than 
many  panes,  and  this  knowledge  is  so  far  controlling,  that  the  sight 
of  the  house  affords  a  mental  impression  of  an  object  much  larger 
than  the  pane,  though  the  merely  sense  impression  is,  that  it  is 
smaller. 

In  relation  to  the  distinction  here  explained,  a  completely  natural 
artist  is  one,  who  sees  things,  only,  and  just  as  his  sense  of  sight 
sees,  without  any  interference  from  thought  or  knowledge  of  real 
relative  sizes  ;  and  who  draws  objects  just  as  his  eye  sees  them. 

Such  a  one  will  spontaneously  conform  to  the  principles  of  per 
spective,  which,  in  relation  to  him,  will  only  be  the  natural  history 
of  his  natural  performances.  In  proportion,  however,  as  knowledge 
of  the  real  sizes  of  objects  warps  the  judgment,  as  to  their  real 
apparent  size  to  the  eye  alone,  does  a  scientific  knowledge  and 
practice  of  perspective  become  necessary  as  a  guard  against  errors 
in  drawing. 

f.  According  to  (14-16)  and  all  the  preceding  constructions,  a 
perspective  drawing  should  be  viewed  from  the  precise  point  from 
which  the  object  represented  is  supposed  to  be  viewed.  Thus,  Fig. 
47  should  be  viewed  by  the  eye  placed  in  a  perpendicular  to  the 
paper  at  E,  and  five  inches  (  •=  CD)  from  that  point.  The  per 
spective  will  then  make  identically  the  same  image  on  the  retina 
that  would  be  made  by  the  original  object  in  its  full  size,  and  25  ft, 
(the  distance  by  scale)  from  n,  on  the  perpendicular  AnC. 


GENERAL    PRINCIPLES   AND   ILLUSTRATIONS.  89 

In  a  picture,  properly  so  called,  where  the  sensible  effect  ia 
greatly  assisted  by  shade  and  color,  if  it  be  viewed  through  a  tube, 
so  as  to  exclude  the  surrounding  objects  which  warp  the  judgment 
when  compared  with  the  small  size  of  the  picture,  the  illusion  may 
be  made  complete,  by  abandoning  the  mind  to  the  picture  exclu 
sively,  and  we  really  seem  to  look  up  through  extended  valleys, 
winding  among  great  hills,  and  overhung  by  a  real  far  distant  sky. 

g.  The  principal  exception  to  this  rule  for  the  position  of  the  eye, 
is  in  viewing  decorative  wall  paintings  of  interiors,  which  may  be 
painted  as  if  seen  from  a  great  distance,  or  otherwise  modified  so  as 
not  to  be  offensively  distorted  to  beholders  in  any  ordinary  position 
within  the  building. 

h.  In  connexion  with  oblique  vision  of  an  object,  as  mentioned 
in  (c),  the  question  occurs,  "  to  what  extent  is  such  vision  admissi- 
ble.'' In  other  words,  what  is  the  practical  limit  of  the  visual  angle. 
We  can  examine  objects  with  the  greatest  minuteness  only  a  point 
at  a  time  or  in  the  line  of  but  a  single  visual  ray  at  a  time.  On  the 
other  hand,  we  can  be  conscious  of  the  existence  of  objects  within 
a  range  of  180°,  either  vertically  or  horizontally.  Where,  now, 
between  these  limits,  is  the  greatest  visual  angle  which  will  allow 
of  a  clear  and  pleasing  general  effect  ?  It  is  usually  supposed  to 
vary  from  45°  to  60°. 

Accordingly,  in  Fig.  47,  by  laying  off  five  inches,  =  CD,  in  front 
of  w,  to  obtain  the  station  point  (113)  (115),  and  from  this  point 
drawing  lines  to  P  and  E,  it  will  appear  that  a  small  visual  angle  is 
formed.  Hence  when  Fig.  47  is  viewed,  as  directed  in  (/),  it  will 
be  very  clearly  seen. 

i.  This  clear  view  is  also  due  to  looking  directly,  in  the  line  CA, 
at  the  centre  of  the  object.  Thus  Fig.  47  is  much  more  satisfactory 
than  Figs.  16  and  17,  PART  I.,  where  the  eye  is  placed  considerably 
to  one  side  of  the  given  object,  partly  to  avoid  the  confounding  of 
plans  and  perspectives,  and  partly  to  avoid  the  very  acute  intersec- 
tions of  lines  of  construction  that  would  have  occurred  had  the 
point  EE'  been  placed  directly  in  front  of  the  objects. 

The  last  consideration  points  to  another  disadvantage  of  the 
method  of  visual  rays^  especially  as  employed  in  connection  with 
two  planes  only. 

EXAMPLE  5. — To  find  the  Perspective  of  a  Cross  and 
Pedestal. 

This  problem  is  chosen  as  one  embracing  numerous  lines  arranged 
in  parallel  sets. 

In  Fig.   49,   let  ABD  be  the  plan  of  the  pedestal,  EFG,  of 


90 


LINEAR    PERSPECTIVE. 


the  horizontal  arm  of  the  cross,  and    HIK,  that  of  its  vertical 
arm. 

LL  is  the  ground  line  which  indicates  the  first,  and  L'L'  the  one 
which  indicates  the  second  position  of  the  perspective  plane.  VV 
is  the  level  of  the  eye,  and  therefore  by  (105)  contains  the  vanishing 
points  of  all  the  horizontal  lines  of  the  object.  S  is  the  station 
point  (113)  taken  in  a  perpendicular  to  the  perspective  plane 
through  the  centre  of  the  object  (I22c). 


-  m,' 


FIG.  49. 

Then,  drawing  SL,  parallel  to  AB,  and  LV  perpendicular  to  LI^ 
we  find  V,  the  vanishing  point  of  all  lines  in  the  direction  of  AB. 
In  a  similar  manner  V  is  found.  Other  points  in  the  indefinite 
perspectives  of  the  horizontal  lines,  are  where  those  lines  pierce  the 
perspective  plane. 

Accordingly,  as  'shown  by  the  figure,  and  (I22a)  and  assuming 
ca  as  the  height  of  the  pedestal,  MB,  produced,  meets  the  perspec- 
tive plane  at  m',  where  m'm"  =  ac  /  and  BA  meets  the  perspective 
plane  at  a.  Then  ra'V  is  the  indefinite  perspective  of  BM,  and 
aV,  that  of  AB.  Hence  #,  their  intersection,  is  the  perspective  of 
B.  From  #,  draw  be,  perpendicular  to  the  ground  line  and  limited 


GENERAL   PRINCIPLES    AND   ILLUSTRATIONS.  91 

by  cV,  and  one  face  of  the  pedestal  will  be  represented.  The  con- 
struction of  its  other  visible  surfaces  is  similar  to  the  foregoing,  as 
is  seen  in  the  figure. 

To  find  the  foot  of  the  vertical  arm.  Kp  and  K^  pierce  the  per- 
spective plane  at  p'  and  <?',  at  heights  equal  to  ac.  p'V  and  q'V 
are  their  perspectives,  which  intersect  at  the  perspective  of  K  in 
the  foot  of  this  arm.  Other  like  points  are  similarly  found.  On 
either  of  the  same  perpendiculars  through  p  and  g,  set  off  the  height, 
as  at  r,  of  the  whole  cross  and  pedestal,  then  rV,  the  perspective  of 
pK  in  the  level  of  the  head  of  the  arm,  will  meet  the  right  hand 
edge  at  &,  the  perspective  of  the  top  point,  K. 

To  find  points  in  the  perspective  of  the  short  arm  EG.  Produce 
its  horizontal  edges  to  the  ground  line,  as  at  q  and  n.  Set  off,  as 
before,  the  heights  of  these  points,  as  at  q"  and  nf  in  the  plane  of  the 
top  of  this  arm.  Then  g,  for  example,  the  intersection  of  <?"V  and 
w'V,  will  be  the  perspective  of  G.  Other  points  may  be  similarly 
found. 

Remarks. — a.  It  is  evident  from  this  problem  that  the  method 
by  horizontal  lines  of  the  object,  when  such  exist,  is  as  convenient 
in  respect  to  avoiding  the  necessity  of  full  vertical  projections,  as  is 
the  method  of  diagonals  and  perpendiculars. 

b.  The  intersection  at  #,  for  example,  is  very  acute,  but  would 
have  been  less   so   had  a  perpendicular,  through  B,  been  used, 
together  with  aV.    Thus,  by  an  adaptation  of  auxiliary  lines  to  the 
conditions  of  each  point,  we  can  obtain  the  perspective  of  each  by  a 
well  defined  intersection. 

c.  The  passing  of  SL  through  e,  and  of  SY  through  d,  are  merely 
accidental  coincidences,  which  are  always  liable  to  occur,  but  which 
never  need  perplex  the  draftsman  if  he  will  retain  a  clear  view  of 
first  principles,  and  keep  in  mind  what  each  line  of  the  figure  really 
means* 


D*  LINEAR   PERSPECTIVE. 


CHAPTER  H. 

PERSPECTIVES   OF   SHADOWS. 

123.  Returning  to  Fig.  35,  PART  I.,  where  BR  represents  a  ray  of 
light  and  AR,  its  horizontal  projection  (96),  it  is  evident  that  R,  the 
shadow  of  B  on  the  horizontal  plane,  is  the  intersection  of  the  ray 
through  B,  with  its  horizontal  projection. 

It  follows,  now,  that  the  perspective  of  R  will  be  the  intersection 
of  the  perspective  of  the  ray  with  the  perspective  of  its  horizontal 
projection  (117). 

Ifj  then,  we  can  find  these  latter  lines,  we  can  find  the  perspective 
of  the  shadow  directly,  or  without  finding  its  projections. 

124.  But  rays  of  light,  proceeding,  as  usually  supposed,  from  tho 
sun,  are  parallel ;  hence  their  vanishing  point,  like  that  of  other 
parallels,  is  found  by  determining  where  a  parallel  ray  of  light 
through  the  point  of  sight  pierces  the  perspective  plane  (108).     Also, 
rays  being  parallel,  their  projections  on  either  plane  will  be  parallel 
(53).     Their   vertical   projections,  being   lines   in   the  perspective 
plane,  will  be  their  own  perspectives ;  and  their  horizontal  projec- 
tions, being  horizontal  lines,  will  have  a  vanishing  point  in  the  hori- 
zon (105).     Hence  the  perspectives  of  rays,  and  of  their  horizontal 
projections,  can  be  found  ;  and  therefore  perspectives  of  shadows  on 
the  horizontal  plane,  can  be  found  directly,  or  without  previously 
finding  the  projections  of  those  shadows,  as  in  PART  I. 

The  principle  just  established,  combined  with  other  general  ele- 
mentary principles  already  employed,  will  also,  as  will  soon  appear, 
3nable  us  to  find,  directly,  all  ordinary  shadows. 

We  proceed  next  to  illustrate  the  principles  just  explained,  in 
some  elementary  constructions. 

EXAMPLE  6. — To  find  the  vanishing  point  of  given  Rays, 
and  of  their  Horizontal  Projections. 

Let  R  and  R',  Fig.  50,  be  the  two  projections  of  a  ray  of  light, 
which  lies  in  front  of  the  perspective  plane,  GL  being  the  ground 
tine.  Also,  let  EE'  be  the  point  of  sight.  Then  EA,  parallel  to  R, 
and  E'V,  parallel  to  R',  are  the  projections  of  a  visual  ray  parallel 
to  the  rays  of  light.  This  visual  ray  meets  the  perspective  plane  in 


PERSPECTIVES    OF   SHADOWS.  93 

the  point  whose  horizontal  projection  is  h,  and  whose  vertical  pro- 
jection, or  the  point  itself,  is  V  (58).  Hence  V  is  the  vanishing 
point  of  all  rays  of  light  parallel  to  R-R'. 

Again,  remembering  that  the 

ft  .  vertical  projection  of  a  hori- 

zontal line,  oblique  to  the  ver- 
tical plane,  is  a  line  parallel  to 
the  ground  line  (51)  E/i,  again, 
U  parallel  to  R,  and  E'H,  paral- 
lel to  GL,  are  the  projections 
of  a  visual  ray,  parallel  to  the 
horizontal  projections  of  the 
rays  of  light.  This  visual  ray 
pierces  the  vertical  or  perspec- 
tive plane  at  H,  which  is  there- 
fore the  vanishing  point  of 
horizontal  projections  of  rays,  and  is  in  the  horizon  E'H. 

125.  Having  completed  Fig.  50,  observe  that  H  and  V,  are,  by 
construction,  necessarily  in  the  same  perpendicular  to  the  ground 
line.     This  follows  from  the  fact  that  a  ray,  and  its  horizontal  pro- 
jection, are  in  the  same  vertical  plane  (81),  and  therefore  the  visual 
rays,  as  E^-E'V,  and  EA-E'H,  parallel  to  them,   are  in  a  vertical 
plane  ;  while  the  vertical  trace  of  such  a  plane,  in  which  these  visual 
rays  pierce  the  vertical  plane  of  projection,  is  a  line  perpendicular 
to  the  ground  line  (79). 

126.  In  general,  if  a  ray,  or  any  other  line,  be  contained  in  a  cer- 
tain plane,  it  must  pierce  any  surface  in  the  intersection  of  the  plane 
with  that  surface  ;  that  is,  in  the  trace  of  the  plane  upon  that  sur- 
face. 

127.  When  a  particular  direction  of  the  light  is  given,  as  at 
R-R',  H  and  Y  must  be  constructed  by  (Ex.  6),  but  if,  as  is  usual  in 
general   problems,  its  direction  is  not.  given,  H   and  V   may  be 
assumed,  agreeably  to  (125). 

EXAMPLE  7. — To  find  the  Perspective  of  the  Shadow  ot 
any  Vertical  Line  upon  the  Horizontal  Plane. 

Let  B'L,  Fig.  51,  be  the  ground  line  ;  which,  in  figures  of  so  few 
lines  as  Figs.  50  and  51,  and  similar  ones,  need  not  be  translated. 
Let  A  be  the  horizontal,  and  A'B'  the  vertical  projection  of  the 
given  vertical  line.  Its  perspective,  db,  is  found  by  visual  rays,  as 
AE — A'E',  as  explained  in  PAKT  I.,  EE'  being  the  point  of  sight. 

Then,  by  (125)  assume  V  and  H,  as  the  vanishing  points  of  rays 
and  horizontal  projections  of  rays,  respectively.  Then  aV  is  the 


94 


LINEAR   PERSPECTIVE. 


perspective  of  the  ray  of  light  through  the  point  whose  perspec- 
tive is  a,  that  is  through  A  A',  and  5H  is  the  perspective  of  the 


B' 


\ 


FIG.  51. 

horizontal  projection  of  the  same  ray.  Hence  by  (123)  II  is  the 
perspective  of  the  shadow  of  a  upon  the  horizontal  plane  ;  and  &R 
is  the  perspective  shadow  of  ab  on  the  same  plane. 

Remarks. — a.  Remembering  that  when  the  rays  are  parallel,  as 
here  supposed,  they,  and  their  horizontal  projections,  will,  each, 
have  a  common  vanishing  point,  it  is  evident,  that  if  the  shadows 
of  a  number  of  vertical  lines,  like  ob,  be  found  on  the  same  hori- 
zontal plane,  they  will  all  converge  to  the  point  H.  The  student 
should  make  this  construction. 

b.  When  the  vanishing  point  Y  is  below  the  horizon,  and  the 
light  proceeds  as  indicated  by  the  arrow  in  Fig.  51,  it  shows  that 
the  light  proceeds  from  above  and  behind  the  left  shoulder.     If  V 
were  above  H,  it  would  indicate  that  the  light  proceeded  from 
above  and  in  front  of  the  right  shoulder  ;  and  the  shadows  would 
fall  towards  the  observer  and  to  the  left,  as  will  readily  appear  on 
making  the  construction. 

c.  If  the  source  of  light  were  a  near  point,  as  a  candle,  the  per- 
spective  of  this  point  and  of  its  horizontal  projection  must  be 
found.     Lines  from  the  former  to  points  in  the  perspective  object 
will  be  the  perspectives  of  rays ;  and  lines  from  the  latter  point 
to  the  perspectives  of  horizontal  projections  of  the  same  points  of 
the  object,  will  be  the  perspectives  of  horizontal  projections  of  rays. 


PERSPECTIVES    OF    SHADOWS.  9-*> 

Rays  from  such  a  luminous  point  diverge  in  every  direction, 
hence  if  in  Fig.  51,  we  suppose  AA'  to  be  the  luminous  point,  a  is 
its  perspective ;  and,  not  aV  only,  but  any  line  from  a  will  be  the 
perspective  of  some  ray. 

The  completion  of  the  construction  thus  far  explained,  is  left  as 
an  exercise  lor  the  student 


LTNKAK    PERSPECTIVE. 


CHAPTER 


MISCELLANEOUS   PROBLEMS. 

128.  The  following  problems  are  added,  not  to  illustrate  any  ne~W 
principles,  but  to  familiarize  the  student  more  fully  with  the  appli- 
cation  of  those  already  explained,  to  practical  problems. 

Premising  that  the  drawing  of  exterior  and  interior  views  of 
buildings,  with  their  accompaniments ;  arcades,  pavements,  and 
furniture,  is  perhaps  the  chief  exact  application  of  perspective,  this 
chapter  is  occupied  with  examples  of  this  character,  the  execution 
of  which  will  enable  the  draftsman  to  proceed  with  the  perspective 
drawing  of  Avenues,  Bridges,  Machines,  etc.,  and  with  the  correct 
additions  of  features  of  natural  scenery  to  the  geometrical  portions 
of  his  drawings. 

EXAMPLE  8. — To  find  the  Perspective  of  a  Pavement  of 
Squares,  whose  sides  are  parallel  to  the  ground  line. 

Let  GB,  Fig.  52,  be  the  ground  line,  DC  the  horizontal  line,  or 
horizon,  and  AGK  a  group  of  twelve  squares,  lying  in  the  horizon- 
tal plane,  and  with  one  side,  GK,  taken  as  the  ground  line. 


Operating  by  the  method  of  diagonals  and  perpendiculars,  let  C  be 
the  centre  of  the  picture  (113)  and  D  the  vanishing  point  of  diago- 
nals. C  is  the  vanishing  point  of  perpendiculars  (112)  and  these 
perpendiculars,  as  LK,  AH,  etc.,  pierce  the  perspective  plane  at  K, 
H,  etc.,  hence  (116)  KC,  HC,  etc.,  are  their  perspectives.  AB  is 


MISCELLANEOUS   PROBLEMS. 


97 


the  diagonal  from  A  and  BD  is  its  perspective.  Therefore  a  is  the 
perspective  of  A.  Likewise  e  is  the  perspective  of  E,  and  /,  of  F. 
Drawing  parallels  to  GK,  through  a,  e,  and/,  (111)  they  will  inter 


FIG.  53. 

feect  the  perspectives  of  the  perpendiculars,  so  as  to  complete  the 
required  perspective  GmlK. 

7 


9S  LINEAR   PERSPECTIVE 

EXAMPLE  9. — To  find  the  Perspective  of  a  Pavement  of 
Hexagons,  -whose  sides  make  angles  of  30°  and  90°  with 
the  ground  line. 

Let  HE  be  the  ground  line,  Fig.  53.  Construct  an  equilateral 
triangle,  as  ABD,  with  one  of  its  sides  perpendicular  to  the  ground 
line.  Divide  either  of  its  sides,  as  AB,  into  any  convenient  num 
ber  of  equal  parts.  Through  each  of  the  points  of  division,  as  Q, 
draw  indefinite  lines,  as  QN  and  QP,  parallel  to  the  remaining 
sides,  DB  arid  DA,  of  the  triangle. 

Portions  of  these  lines,  together  with  perpendiculars,  as  OP, 
joining  the  proper  intersections,  which  will  be  obvious  on  inspection, 
will  form  a  group  of  regular  hexagons.  These  may  be  limited  at 
pleasure,  as  by  the  rectangle  HEFG. 

Now  let  LL  be  the  ground  line,  after  translation,  CV  the  hori- 
zontal line,  C  the  centre  of  the  picture,  and  S  the  station  point, 
taken  in  this  case,  for  variety,  at  one  side  of  the  middle  of  the 
figure. 

The  sides  of  the  hexagons,  forming  parallel  groups,  are  taken 
as  lines  of  construction.  Their  vanishing  points — beyond  the  limits 
of  the  figure — are  found  by  drawing  lines  at  S  (partly  shown) 
parallel  to  HK  and  IG,  till  they  meet  HE.  From  the  latter  points^ 
perpendiculars  to  CV  produced,  will  meet  CV  in  the  vanishing 
points  of  HK  and  IG  and  of  all  lines  parallel  to  them.  The  re- 
maining lines  of  the  hexagons  are  perpendiculars,  and  C  is  their 
vanishing  point. 

Observing  that  the  plans  and  perspectives  of  the  same  points 
have  the  same  letters,  the  remainder  of  the  construction  needs  no 
further  explanation. 

HemarJc.  If,  in  Ex.  8,  the  squares  had  been  placed  with  their 
sides  making  angles  of  45°  with  the  ground  line,  those  sides  would 
all  have  been  diagonals,  instead  of  parallels  and  perpendiculars. 

In  the  last  example,  if  AB  had  been  taken  in  the  ground  line,  c 
he  sides  of  the  hexagons  would  have  made  angles  of  60°  with '' 
he  ground  line,  except  those  which  would  have  been  parallel  to  it. 
Hence,  SR  remaining  the  same,  the  vanishing  points  of  the  inclin- 
ed sides  would  have  been  nearer  to  C.       The  student  should   re- 
construct these  examples  under  these  new  conditions. 

EXAMPLE  10. — To  find  the  Perspective  of  an  Interior. 

Preliminary  explanation.  According  to  (12  2 A)  a  person  stand- 
ing against  one  wall  of  a  room,  can  be  conscious  of  the  entire  in- 
terior, though  the  whole  cannot  be  distinctly  recognized.  If,  then, 
Fig.  54,  a  person  stand  at  E,  seeing  clearly  everything  within  the 


MISCELLANEOUS    PROBLEMS.  99 

visual  angle  AEB,  only  the  portion  of  the  room  be 
yond  AB  can  be  represented  in  a  picture.     Hence, 
A  if  a  larger  portion,   or  the  whole  of  the  interior  is 
,  to  be  represented,  the  near  wall  A^B"  must  be  sup 
posed  to  be  removed,  so  that  E',  or  E",   may  be  the 
position  of  the  observer,  from  which  all  beyond  A'B', 
or  A"B",  will  be  visible. 

Construction  of  Fig.   55.     In   this   example,    let 
the  near  wall  be  removed,  and  let  the  whole  interior 
be  seen  under  a  visual  angle    of   45°.      ABGL   is 
the  plan  of  the  room,    with   an   elliptically  arched 
passage,  of  the  width  EF,  opening  out  of  it  on  the  right,  and  with 
a  door,  HK,  in  the  left  wall. 

Let  the  observer  stand  opposite  the  point  X,  at  one  third  the 
width  of  the  room  from  G.  We  have  then  to  construct  S,  the  ver- 
tex of  an  angle  of  45°,  whose  base  is  GL,  and  placed  opposite  to 
X.  Draw  GT  and  LT,  each  making  an  angle  of  45°  with  GL. 
Draw,  at  X,  a  perpendicular  to  GL,  and  with  T  as  a  centre  and 
TG  as  a  radius,  describe  a  small  arc,  intersecting  this  perpendicular 
at  S,  which  will  be  the  station  point  as  required. 

Now  let  G'L'  be  the  ground  line,  indicating  the  second  position  of 
the  perspective  plane,  and  let  CD  be  the  horizontal  line.  This 
line  must,  if  the  observer  is  supposed  to  stand  on  the  floor,  be 
about  five  feet  above  G'L',  on  the  same  scale  on  which  the  plan, 
AG,  is  drawn.  Note  that  C  is  in  the  perpendicular  XS,  produced 
to  meet  CD. 

Observing,  now,  that  a  diagonal  from  A  will  meet  LG  at  a  dis- 
tance to  the  left  of  L,  equal  to  LA,  and  so  for  other  points,  the 
diagonals  themselves  need  not  be  drawn.  Thus,  make  CD  =  SX 
(115)  and  D  will  be  a  vanishing  point  of  diagonals.  Then  make 
L'A  ==LA  and  A'D  will  be  the  perspective  of  the  diagonal  from 
4.  The  perspective  of  the  perpendicular  LA  is  L'C,  hence  a  is  the 
perspective  of  the  right  hand  back  corner,  A,  of  the  floor.  Draw 
«6,  parallel  to  L'G',  till  it  meets  G'C,  the  perspective  of  GB,  and 
L'G'  db  will  be  the  perspective  of  the  floor. 

The  front  wall  of  the  room,  GL,  being  taken  as  the  perspective 
plane,  the  intersection  of  the  room  with  that  plane  will  be  its  own 
perspective,  in  full  size  and  real  form.  Hence  make  L'L"  and  G'G" 
equal  to  the  height  of  the  walls  \  and,  supposing  the  coiling  to  be 
semicircular,  describe  a  semicircle  on  L"G"  as  a  diameter. 

As  an  example  of  a  simple  cornice,  in  perspective,  make  the 
small  rectangles  at  L"  and  G",  as  sections  of  it  in  the  perspective 


100 


LINEAR  PERSPECTIVE. 


MISCELLANEOUS   PKOBLEMb.  101 

plane.  Then  draw  its  edges  towards  C,  limiting  it  by  a  Horizontal 
and  vertical  line  where  its  lower  back  edge,  on  each  wall,  meets 
the  vertical  lines  from  a  and  b. 

QC  is  the  perspective  of  a  perpendicular  through  the  centre  of 
the  floor.  Hence  qs  is  the  perspective  of  the  centre  line  of  the  fur- 
ther wall.  Where  qs  meets  a's,  a'  being  the  intersection  of  aar 
nd  L"C,  is  the  centre  of  the  semicircular  boundary  of  the  further 
end  of  the  ceiling ;  which  is  a  semicircle  in  perspective,  because  it 
is  parallel  to  the  perspective  plane.  If  there  be  a  round  topped 
window  in  the  centre  of  the  further  wall,  lay  off  its  half  width, 
QR=QI,  each  side  of  the  middle  point  Q,  draw  RC  and  1C,  and  per- 
pendiculars to  ab  as  at  r.  Then  make  L'v  equal  to  the  height  of 
the  base  above  the  floor,  draw  vC,  and  v'r'  paralled  to  ab,  and 
the  semicircular  top,  with  s  as  a  centre,  to  have  it  concentric  with  the 
ceiling.  This  will  complete  the  outline  of  the  window. 

To  draw  the  opening  HK  (which  is  very  wide  in  order  to  show 
the  construction  more  plainly).  Draw  HS  and  KS,  horizontal  pro- 
jections of  visual  rays,  or  horizontal  traces  of  vertical  visual  planes, 
through  the  sides  of  the  opening.  Then  the  intersections  of  these 
planes  with  the  perspective  plane,  at  hw  and  k,  drawn  from  H'  and 
K',  will  be  the  perspectives  of  the  vertical  doorway  lines  at  II  and 
K.  Make  G'W  equal  to  the  height  of  the  door,  and  WC  will 
limit  the  inside  of  the  top  of  the  door.  Next  draw  the  edges  in 
the  thickness  of  the  doorway  as  H,  parallel  to  G'L',  the  vertical 
line  It,  and  from  t  the  line  towards  C,  which  completes  the  doorway. 

To  draw  the  archway  EF.  Make  L'E'=LE,  and  L'F'=LF. 
Draw  E'D  and  F'D,  which  will  give  e,  and/*,  the  perspectives  of  E 
and  F.  Make  I/O"  equal  to  the  height  of  the  vertical  portion  of 
the  archway,  and  limit  the  vertical  lines  at  e  and  f,  by  O"C.  To 
find  the  perspective  of  the  highest  point,  draw  the  semi-ellipse, 
FPE,  representing  the  elliptical  top  of  the  arch  as  revolved  round 
EF,  till  parallel  to  the  horizontal  plane.  Lines,  as  OP,  in  this 
semi-ellipse,  are  called  ordinates.  Take  the  longest  ordinate,  OP, 
set  it  off  at  O"P"  and  draw  P"C.  Make  L'O'=LO,  draw  O'D, 
and  op,  then  p  will  be  the  perspective  of  O,  that  is  of  P.  Any 
point  in  the  ellipse  may  be  similarly  found.  Thus,  take  MN  any 
where,  and  parallel  to  OP.  Make  O"N'=MN,  and  draw  N'C. 
Also  make  L'M'=LM,  draw  M'D  and  mn,  then  n,  the  intersection 
of  mn  with  N'C,  will  be  the  perspective  of  N.  After  finding  one 
or  two  more  points  in  like  manner,  the  perspective  ellipse,  fpne' 
can  be  sketched.  The  horizontal  line  at  /will  then  complete  the 
archway,  and  the  whole  figure. 


102 


LINEAR    PERSPECTIVE. 


EXAMPLE  11. — To  find  the  Perspectives  of  the  Shadows 
in  an  Interior. 

In  order  not  to  confuse  figure  55,  the  constructions  of  the  re- 
quired shadows  are  made  on  the  following  enlarged  copies  oi 
detached  portions. 

First  y  to  find  the  shadow  of  the  edges  of  the  doorway,  Fig.  56. 
Supposing  no  particular  direction  of  the  light  to  be  given,  assume 
H  as  the  vanishing  point  of  horizontal  projections  of  rays,  and  R, 


FIG.  56. 

as  the  vanishing  point  of  rays.  It  is  readily  apparent  on  considera 
tion,  that  ka,  tc  and  cd  are  those  edges  of  the  doorway,  parts  of 
which,  at  least,  will  cast  shadows.  MI  is  the  perspective  of  the 
horizontal  projection  of  all  rays  through  the  vertical  edge  Tea.  That 
is,  it  is  the  horizontal  trace  of  a  vertical  plane  of  rays  (99)  through 
Jca.  It  is  therefore  the  shadow  of  lea  on  the  floor,  as  far  as  m, 
where  it  meets  the  further  wall  ABD.  Thence,  this  plane  being 
vertical,  its  trace,  and  shadow  of  ka  on  ABD,  is  vertical,  as  seen  at 
mE. 

To  find  E,  consider  that  ct  will  partly  cast  a  shadow  on  the 
surface  atk.  This  surface  being  parallel  to  the  perspective  plane, 
and  ct  perpendicular  to  it,  the  shadow  of  ct  on  atk  will  be  parallel 


MISCELLANEOUS    PROBLEMS. 


10b 


to  the  vertical  projection,  CR,  of  a  ray  of  light  (97),  and  will  begin 
at  £,  where  ct  meets  atk.  Hence  fe,  parallel  to  CR  (HI)  is  the 
shadow  of  ct  on  atk.  Therefore  e  is  the  highest  point  of  ha  that 
can  cast  a  shadow.  Hence  draw  the  ray  eR,  and  E,  its  intersection 
with  mE,  will  be  the  limit  of  the  shadow  of  ka.  The  remainder 
of  the  construction  is  now  evident  from  the  figure. 

Since  light  streams  through  the  doorway,  the  area  within  th 
shadow  of  its  edges  is  light,  as  indicated  by  the  partial  shade  line 
of  the  figure. 

Second.  To  find  the  shadow  in  the  archway,  Fig.  57.  This 
shadow  is  in  four  parts ;  first,  the  shadow  of  the  edge  ee'  upon  the 
floor ;  second,  that  of  the  same  edge  on  the  wall  Qff  ;  third,  that 
of  the  curve  e'pf  on  the  same  wall ;  and  fourth,  that  of  the  same 
curve  upon  the  cylindrical  surface  of  the  archway,  above  the  hori- 
zontal plane  through  e'  and/'. 

Let  H  be  the  vanishing  point  of  horizontal  projections  of  rays, 
and  R,  the  vanishing  point  of  rays.  Then  eR  is  the  perspective  of 


Cu~,,, 


FIG.  57. 

the  horizontal  trace  of  a  vertical  plane  of  rays  through  ee\  and  eG 
is  the  shadow  of  this  edge  on  the  floor.  By  drawing  the  ray  RG, 
and  producing  it  to  g,  we  learn  that  eg  is  the  precise  portion  of  ee' 
which  casts  a  shadow  on  the  floor.  From  G,  the  shadow  of  ge1  is 


104  LINEAR   PERSPECTIVE. 

the  vertical  line  GE,  limited  at  E  by  the. ray  e'R.  Above  E,  the 
shadow  is  cast  by  the  arch  curve,  and  is  found  as  follows.  Assume 
any  point,  q\  and  draw  the  vertical  line  q'q,  which  is  the  trace  of  a 
vertical  plane  of  rays  through  q\  upon  the  side  of  the  room.  Then 
qH  is  the  perspective  of  the  trace  of  this  plane  upon  the  floor,  and 
the  vertical  line  from  the  intersection  of  qH.  with  /"G,  is  its  trace  on 
the  wall  of  the  arch.  This  plane  contains  the  ray  #'R,  which  meets 
the  vertical  line,  just  named,  at  Q,  which  is  therefore  the  shadow  01 
q'.  D,  the  shadow  of  d'  is  found  in  like  manner.  T,  the  point  of  con- 
tact  of  the  ray  TR,  with  the  arch  curve/^?e',  is  the  upper  end  of  the 
shadow,  which  may  be  sketched  by  joining  the  points  already  found. 

In  this  figure,  the  shadow  on  the  cylindrical  surface  of  the  arch 
is  so  small,  that  no  points  in  it  have  been  found,  except  T.  The 
most  elementary  method  of  finding  points  of  this  shadow,  is  the 
following  indirect  one,  which  is  so  fully  indicated  that  the  student 
will  probably  find  no  difficulty  in  applying  it  for  himself.  Assume 
any  point  as  h  quite  near  to  /",  on  I/a,  and  draw  through  it  lines 
parallel  to  /G  arid  ff,  which  will  represent  a  plane  parallel  to  the 
wall  Off.  This  plane  will  cut  a  line  from  the  arch,  parallel  to  /G, 
and  beginning  where  the  vertical  line  from  h  meets  the  arch  curve. 
Next  find  a  few  points  of  shadow  on  this  plane,  just  as  DQE  was 
found.  Then  the  intersection  of  this  auxiliary  shadow  with  the 
horizontal  line  cut  from  the  arch,  will  be  a  point  of  shadow  on  the 
arch  (99)  and  by  drawing  a  ray  from  R  through  this  point,  we  can 
find  the  precise  point  onfpe'  which  casts  this  point  of  shadow. 

EXAMPLE  12. — To  find  the  Perspective  of  a  Cabin. 

In  this  example,  a  variety  of  methods  will  be  employed,  by  way 
of  review ;  also  some  special  operations,  suited  to  the  construction 
of  particular  points. 

Let  ABD,  Fig.  58,  be  the  plan  of  the  cabin  walls,  EF  of  its  roof 
ridge,  and  H"HI  of  Its  chimney.  Let  the  perspective  plane  be 
taken  at  GK,  through  the  corner  A,  and  let  G'K'  be  its  ground 
line  after  translation  and  revolution  into  the  plane  of  the  paper. 
Let  W  be  the  horizon,  C  the  centre  of  the  picture,  and  S  the 
station  point.  The  perpendicular  to  the  ground  line,  and  contain- 
ing S  and  C  passes  through  *,  the  centre  of  the  plan  (122  i). 

The  edge  at  A,  being  in  the  perspective  plane,  is  its  own  perspec- 
tive, and  appears  in  its  real  height  at  aa'.  The  visual  ray  BS — 
L'C  pierces  the  perspective  plane  at  #,  the  perspective  of  the  lower 
corner  at  B.  Make  L"B"=aa',  then  BS — B"C  is  the  visual  ray 
from  the  upper  corner  at  B,  and  b'  is  the  perspective  of  that  corner. 
Draw  db  and  a'b'. 


MISCELLANEOUS    PROBLEMS. 


10ft 


FIG.  88. 


106  LINEAR   PERSPECTIVE. 

The  vanishing  point  of  all  lines  parallel  to  AB,  can  now  be  found 
in  either  of  two  ways.  In  the  usual  way,  it  would  be  found  by 
drawing  through  S  a  line  parallel  to  AB,  till  it  meets  GK,  whence 
drop  a  perpendicular  to  VV  (108).  Or,  produce  db  and  a'b'  till  they 
meet  VV,  in  the  same  vanishing  point ;  which,  being  out  of  the 
paper,  is  indicated  by  V". 

Likewise  find  V,  the  vanishing  point  of  all  lines  parallel  to  AD, 
in  the  usual  way,  if  before  finding  dd,  or  as  just  explained,  if  after 
finding  dd',  as  shown  in  the  figure.  Having  found  the  end,  add' , 
of  the  cabin,  the  intersection,  e,  of  its  diagonals  ad  and  da ',  is  the 
perspective  centre  of  that  end,  over  which,  in  the  vertical  line  ee', 
the  peak  of  the  roof  is  found  as  follows.  Lay  off  the  real  height, 
projected  from  E,  at  E"  ;  then  E"C,  the  perspective  of  a  perpen- 
dicular from  E,  will  intersect  ee'  at  e',  the  perspective  of  EE". 
Now  draw  a'e'  and  d'e',  the  perspectives  of  the  left  end  lines  of 
the  roof.  These  lines  are  in  the  same  vertical  plane  with  ad,  hence 
their  vanishing  points  are  in  the  perpendicular,  GG',  to  G'K'  and 
through  V  (125-6).  Hence  produce  a'e'  to  meet  GG'  in  R,  which 
will  j|be  the  vanishing  point  of  all  lines  parallel,  in  space,  to  a'e. 
Also  e'd,  produced  to  T,  makes  T  the  vanishing  point  of  all  lines 
parallel  to  e'd.  To  find  R  and  T  by  the  usual  process,  consider 
that  a'E"  and  D"E"  are  the  vertical  projections  of  AE  and  DE, 
and  then  find  where  lines  through  the  point  of  sight  C,  S,  and 
parallel  to  a'E" — AE  and  D"E" — DE,  pierce  the  perspective  plane, 
which  will  be,  as  before,  at  R  and  T.  Next  draw  e'f  to  V",  and 
b'fto  R,  which  will  complete  the  perspective  of  the  roof. 

To  find  the  perspective  of  the  chimney,  and  first  of  its  base. 
Produce  IH  to  J  and,  drawing  the  visual  ray  JS,  project  J'  into 
the  edge  of  the  roof  at  j.  Then  draw  jh  through  V";  or,  by  ele- 
mentary geometry,  draw  b'f"  parallel  to  a'e',  and  limit  it  by  e'f 
produced,  then  divide  a'e'  and  b'f"  proportionally  at  j  and  f 
and  draw  jf  (For  example,  if  e'j  is  J  of  e'a',  then  make/'/7t/"==^  of 
•/"'#').  Find  h  by  the  visual  ray  HS,  whose  intersection  with  the 
perspective  plane  at  H'  is  projected  into  jf  at  h.  Find  u  in  the 
same  way  from  I.  To  find  i,  set  up  the  full  height  of  the  chimney 
top  from  the  ground  at  i',  projected  from  I,  and  draw  the  perspec- 
tive perpendicular  i'C  to  limit  the  vertical  edge  ui  at  i.  Other 
wise:  (Ex.  5.)  produce  the  right  hand  side  of  the  chimney  to  Y , 
and  set  up  its  height,  projected  from  I",  at  i",  and  limit  ui  by 
t'V.  Then  limit  hh'  by  ih'  drawn  through  V",  and  draw  A'V. 
Draw  AR  until  the  ridge  is  met,  thence  a  line  towards  T,  limited 
as  follows.  Draw/V  and  note/,  its  intersection  with  e'd',  whence 


MISCELLANEOUS   PROBLEMS.  107 

draw  a  line,  j'h",  to  V",  limiting  an  edge  of  the  chimney  at  A", 
whence  draw  this  edge,  which  is  limited  by  h'V. 

In  finding  the  door  and  window,  further  special  constructions 
will  be  used,  as  proposed. 

If  lines  be  drawn,  parallel  to  any  line  as  BK,  AB  and  AK  will 
be  similarly,  that  is,  proportionally  divided,  and  if  AK=AB,  these 
similar  parts  will  be  equal,  and  in  the  same  order.  Hence  make 
«K'=AB,  and  K'#  will  be  the  perspective  of  KB,  and  V",  its  in- 
tersection with  W,  will  be  the  vanishing  point  of  all  parallels  to 
KB,  (106).  Then  make  aP  and  aO  equal  to  the  distances  of  the 
two  sides  of  the  door  from  a,  that  is  from  A,  and  draw  PV"  and 
OV"  which  will  meet  db  at  p  and  o,  the  perspectives  of  the  sides 
of  the  threshold.  Set  off  the  true  height  of  the  door  from  a  on 
aa'  and  draw  a  line  to  V",  which  will  complete  the  door  by  limit- 
ing the  verticals  at  p  and  o. 

In  like  manner,  a  window  in  the  front  of  the  cabin  could  be  put 
in  perspective. 

To  find  the  perspective  of  the  end  window.  According  to  the 
method  just  explained,  make  «G'— AD  (Gf  accidentally  falls  on  the 
perpendicular  RT  (Ex.  5.  Rem.  c.)  and  let  NM  be  the  true  relative 
width  and  place  of  the  window.  Draw  GWV,  analogous  to  K'#, 
also  NV  and  MV.  At  n  and  m  draw  vertical  lines,  and  having 
made  aa"  equal  to  the  height  of  the  window  seat,  limit  them  by 
a"  V.  Make  aQ  equal  to  the  thickness  of  the  cabin  wall,  and  draw 
QV",  noting  q,  its  intersection  with  db.  Then  draw  qq\  limited 
by  a"V",  and  from  q'  draw  <?'V,  which  gives  the  inner  edge  of  the 
window  seat.  From  the  further  upper  and  lower  outer  corners  of 
the  window,  short  lines  are  seen,  which  being  perpendicular  to  the 
end  wall,  are  parallel  to  ab,  and  therefore  vanish  at  V".  The 
lower  one  of  these  lines  is  limited  by  </V,  and  from  the  point 
thus  given,  the  inner  vertical  line  of  the  window  is  drawn,  which 
at  y  limits  the  upper  line  to  V'',  and  the  inner  top  line  which  van- 
shes  at  V. 

The  horizontal  lines  of  the  fence  and  sidewalk  vanish  at  V.  The 
top  of  the  fence  being  in  the  line  VC,  it  is  thus  shown  to  be  about 
five  feet  high.  At  Z  is  shown  a  fragment  of  a  cross  street,  parallel 
to  the  perspective  plane.  The  real  distance  of  this  street  from  a, 
or  A,  is  equal  to  the  distance  from  a  to  the  intersection  of  VZ 
produced  (not  shown)  with  aG'  produced.  The  tree  in  the  yard  is 
seen,  by  comparing  with  the  top  line  of  the  window,  to  be  about 
twelve  feet  high. 

129.  By  comparison  of  Fig.  58,  with  any  of  those  in  PART  I.,  it 


108 


LINEAR    PERSPECTIVE. 


appears,  on  inspection,  that  the  perspective,  as  L'C,  of  aperpendl 
cvlar,  is  the  same  as  the  vertical  projection,  L'C,  of  a  visual  ray, 
BS — L'C,  though  the  same  point.  This  is  evident  from  the 
definitions  of  these  lines.  Each,  it  will  be  seen,  joins  the  vertical 
projection  of  the  point  through  which  it  passes,  with  the  vertical 
projection  of  the  point  of  sight,  which  latter  is  the  centre  of  the 
icture. 

EXAMPLE  13. — To  find  the  Perspective  of  the  Shadow  of 
a  Chimney  on  a  Roof. 

Fig.  59  is  substantially  an  enlarged  copy  of  a  part  of  the  roof  of 
Fig.  58,  but  with  the  proportions,  and  the  level  of  the  eye,  changed, 
merely  to  bring  the  construction  within  the  limits  of  the  page.  H, 


h.' 


FIG.  59. 


in  the  horizontal  line,  is  the  vanishing  point  of  projections  of  rays 
on  any  horizontal  plane  (Ex.  6).     R,  on  HR,  but  not  shown  in  its 
real  position,  is  the  vanishing  point  of  rays. 
Now  to  find  the  shadow  of  the  edge  hh'  upon  the  roof.     This 


MISCELLANEOUS  PROBLEMS.  100 

shadow  will  be  the  trace  on  the  roof,  of  a  vertical  plane  of  rays 
through  AA'.  The  line  ah,  joining  the  intersections  of  the  edges  of 
the  chimney  with  the  roof,  is  horizontal  in  space,  and  in  the  side 
surface  of  the  chimney,  within  the  roof,  cd  is  a  line  in  a  vertical 
plane  through  the  ridge  DC,  and  is  also  in  the  same  side  of  the 
chimney.  Hence  dV '",  drawn  to  V"'  (see  Fig.  58.)  is  the  trace  of 
the  central  vertical  plane,  through  DC,  upon  the  horizontal  plan 
through  ah.  Now  AH  is  the  perspective  01  the  trace  of  a  plane  of 
rays  through  hh'  upon  the  latter  plane ;  en,  a  vertical  line  from  the 
intersection  of  AH  with  <#V",  and  meeting  the  ridge  DC  at  n, 
is  the  trace  of  the  same  plane  of  rays  on  the  vertical  plane 
through  DC.  Hence  nh  is  the  trace  of  the  plane  of  rays 
upon  the  roof.  The  ray  A'R,  in  this  plane,  meets  this  trace 
at  t,  which  is  therefore  the  shadow  of  A'.  Hence  th  is  the  shadow 
of  AA'.  A  portion  of  the  shadow  of  h'b  is  visible,  which  is  found 
as  follows.  Produce  Ac  to  meet  ab  in  r,  which  is  therefore  the  in- 
tersection of  db  with  the  front  side  of  the  roof,  produced.  Next, 
produce  hn  to  T  in  the  perpendicular  RHT,  and  T  will  be  the  vanish- 
ing  point  of  all  traces  of  vertical  planes  of  rays  on  the  front  roof. 
Hence  Tr  is  the  perspective  trace  of  a  vertical  plane  of  rays 
through  ab  upon  the  indefinite  plane  of  the  front  of  the  roof. 
Drawing  the  ray  ftR,  it  gives /as  the  shadow  of  b  upon  the  front 
roof  produced.  Hence  ft  is  the  shadow  of  bh'  on  this  roof,  and 
the  portion,  st,  of  this  shadow  is  real,  and  visible. 

Remark.  In  concluding  these  examples  of  shadows,  and  of  this 
volume,  it  may  be  added,  that  though  few,  the  student  will  find 
them  so  varied,  that,  by  attentively  considering  them,  he  will  doubt- 
less be  able  to  construct  any  ordinary  shadow,  or  at  least  to  judge 
more  accurately  of  the  appearance  of  shadows,  sketched  directly 
or  without  construction. 


110  LINEAR   PERSPECTIVE. 


CHAPTER  IV. 

PICTURES,    AND   AERIAL   PERSPECTIVE. 

130.  FOR  full  instructions  on  the  subject  of  this  chapter,  those 
who   make   perspective   drawings,   primarily  for   pictorial   effect, 
should  consult  books  which  treat  of  perspective  as  an  imitative  art 
(27).      A   few   topics   only  are   here   treated,  principally  for  the 
information  of  those  who  may  have  occasion  to  add  a  few  subordi- 
nate items  of  scenery,  &c.,  to   drawings  mainly  of  a  geometrical 
character. 

131.  Landscape  outlines.     These,  to  be  sketched  .in  their  true 
apparent  position  and  form,  must  be  seen  with  truly  artistic  or 
childlike  sense,  that  is,  without  interference  by  the  knowledge  of 
their  real  position  and  form.     In  other  words,  we  simply  copy 
what  the  eye  sees,  and  just  as -it  sees  it,  and  not  what  the  mind 
infers  from  what  the  eye  sees  (122  e). 

132.  In  proportion  as  we  abandon  reliance  on  simple  sense,  for 
scientific  knowledge,  will  the  unassisted  eye  fail  to  serve  us  per- 
fectly ;  and  some  mechanical  guide  to  it  will  become  necessary  or 
convenient. 

It  would  be  impracticable,  however,  to  make  preliminary  plans 
and  elevations  of  broad  landscape  areas,  such  as  have  been  employed 
in  the  preceding  geometrical  constructions.  Hence  simpler  aids 
are  sought.  Among  these  is  the  use  of  a  pencil  and  string,  as  a 
measure  of  relative  apparent  sizes  and  spaces.  Thus,  if  one  end  of 
a  cord  of  fixed  length  be  held  in  the  teeth,  and  a  pencil,  attached 
to  the  other  end  of  the  cord,  be  always  held  perpendicularly  to  the 
string,  the  latter  being  always  horizontal,  it  will  be  easy  to  measure 
on  the  pencil  the  apparent  dimensions  of  objects,  and  the  distances 
and  directions  of  lines  between  different  prominent  points  in  the 
view  to  be  drawn.  After  this,  the  remaining  outlines  can  be 
sketched  by  the  eye  alone. 

A  more  complete  guide  to  the  draftsman  is  a  frame  of  threads. 
Thus,  by  interposing  at  a  suitable  distance,  that  is,  so  as  to  include 
the  whole  of  a  proposed  view,  a  rectangular  frame,  carrying  threads 
which  cross  each  other  at  right  angles  about  an  inch  apart,  the 


PICTURES,    AND   AERIAL 


actual  landscape  will  be,  to  the  eye,  divided  into  square  inches. 
The  paper  being  then  divided  into  similar  squares,  all  that  is  seen 
in  each  thread  square  can  be  accurately  located,  by  reference  to  ita 
sides,  within  the  corresponding  square  on  the  drawing. 

133.  This  last  process  simplifies  picture  drawing,  by  making  the 
whole  view  consist  of  the  sum  of  many  little  pictures,  each  of 
which  is  so  small,  that  by  fixing  the  undivided  attention  of  the  eya 

pon  it,  it  can  be  drawn  just  as  it  is  seen,  according  to  (131).  By 
keeping  this  in  mind,  and  by  gradually  enlarging  the  squares, 
either  of  the  frame  alone,  or  the  picture  alone,  or  of  both,  the  eye 
may  be  trained  into  independence  of  such  guides. 

134.  Landscape  details.     Trees.     These,  if  few,  large,  and  near, 
and  in  a  real  field,  yard,  or  street,  may  have   their  position   and 
height   indicated   in  projection,  as   in  the  preceding  geometrical 
constructions.     Their   perspectives   will   then    serve   as   guides   in 
sketching  smaller  similar  objects.     Remote  trees  will  be  known  as 
larger  than  near  ones,  if  they  rise  higher  above  the  horizon,  while 
standing  on  the  same  level.     Straight  rows  of  trees  may  be  more 
accurately  drawn,  by  locating  the  vanishing  point  of  the  line  along 
which  they  are  ranged. 

134.  Hills  will  be  known  by  their  rising  above  the  horizon  ;  and 
their  relative  distances,  either  by  the  cutting  off  of  their  crest 
lines  ;  or  by  their  height  above  the  horizon,  if  of  equal  heights  ;  or 
by  their  dimness  of  color  and  shade  when  finished. 

Thus  in  this  little  sketch,  the 
mountain  stream  in  ascending, 
flows  between  the  foremost,  or 
right  hand  hill,  and  the  more 
remote  left  hand  one,  while  the 
dim  central  peak  is  evidently 
distant  and  quite  high. 

136.  Valleys,  below  the  ob- 
server's level,  will  be  known  by 
animals,  shrubs,  <fcc.,  in  them 
appearing  below  the  horizon. 
Also  if  the  descent  into  them  is 
sudden,  they  may  be  plainly  indi- 

cated by  the  crest  of  the  high  ground  before  them,  together  with 
the  greater  distinctness  of  the  objects  shown  in  the  foreground, 
as  in  the  following  sketches  ;  the  first  a  sea  view  from  a  precipitous 
shore,  the  second  a  land  view  of  a  broad  valley  seen  from  near  the 
crest  of  an  elevated  table  land. 


FIG.  60. 


112 


LINEAR   PERSPECTIVE. 


FIG.  61. 


FIG.  62. 

137.  Ascent  and  descent  from  the  observer,  is  indicated,  in  land- 
scape views,  by  the  placing 
of  men,  animals,  shrubs, 
rocks,  &c.,  respectively,  fur- 
ther and  further  above,  or 
below  the  horizon.  la 
street  views,  the  relative  di- 
rection of  the  basement  and 
sidewalk  lines  will  show 
the  same  thing.  Thus  in 
the  engraving,  V  is  the  van- 


FIG.  63. 


ishing  point  of  horizontal 
lines,  and  V,  of  the  street 
Lines  ;  hence  the  view  is  that  of  a  descending  street.  See  also  the 
remote  figure,  whose  eye  is  below  the  horizon. 

In  the  following  figure,  however,  where  V,  the  vanishing  point  of 
the  street  lines,  is  above  V,  that  of  the  horizontal  lines,  the  street 
evidently  ascends. 

138.  Level  of  the  eye.  In  interior  and  street  views,  particularly, 
the  horizon  should  not  be  chosen  thoughtlessly,  or  in  improbable 
positions.  In  Fig.  64,  its  position  indicates  a  view  taken 
either  from  higher  ground  than  that  shown  in  the  figure,  or  from 
the  second  story  of  houses  like  those  on  the  right,  and  standing 
on  the  level  of  the  line  aV. 


PICTURES,    AND    AERIAL   PERSPECTIVE. 


113 


139.  Reflections  in  water.     Two  particulars  may  here  be  ncted. 

First •  reflections  of  the  sun  and  moon  on  the  water  should  not 

make  an  acute  angle  with 
the  horizon,  as  is  some- 
times done,  but  should  be 
W  X  J^4\\\\  perpendicular  to  that  line, 

IIW  '\      flflH*  in  tne  Picture<     TIie  rea- 

slImK/     \         ;«flfWfl  son  is  obvious.     The  inci- 

dent rays  to  the  water, 
and  the  reflected  rays 
from  it  to  the  eye,  by 
which  the  band  of  light  on 
the  water  becomes  visible, 
are  in  a  vertical  plane. 
But  the  intersection  of 
FIG.  64.  this  plane  with  the  per-. 

spective  plane,  is  the  per- 
spective of  this  reflection,  and  this  intersection  is  perpendicular  to  the 
horizon  (79). 

Second /  images  in  the  water  sometimes  show  surfaces  of  an  ob- 
ject, not  seen  directly.     Thus  in  the  annexed  figure  of  a  wall  stand- 


FIG.  65. 


ing  in  water,  the  top  of  the  coping  is  seen  on  the  wall  itself,  but 
the  under  side  in  the  reflection  ;  which,  being  nearest  the  water, 
can  alone  send  rays  to  it  to  be  reflected  to  the  eye. 

140.  Location  of  the  centre  of  the  picture.  This  point,  in  order 
to  afford  an  equally  clear  view  of  all  parts  of  the  picture,  should 
coincide  with  the  geometrical  centre  of  the  drawing.  It  may  be 
varied  slightly  from  this  position,  however,  in  order  to  show  with 
especial  distinctness  the  more  interesting  or  important  parts  of  the 


114  LINEAR   PERSPECTIVE. 

picture.  In  "  bird's  eye  views,"  it  will  be  quite  above  the  middle 
of  the  canvas. 

141.  Location  of  the  perspective,  plane.  This  has,  in  all  the  pre- 
ceding problems,  been  understood  to  be  between  the  eye  and  the 
object.  It  is  not  necessarily  so,  but  is  so  placed  in  order,  first  to 
reduce,  rather  than  expand  any  graphical  errors  in  the  projections, 
and,  second,  to  avoid  the  increased  length  and  confusion  of  the 
lines  of  construction,  which  would  result  from  having  the  perspec- 
tive larger  than  the  projections.  It  is  evident  that  the  latter  con- 
sequence would  follow,  for  the  eye  being  the  vertex  of  the  visual 
cone,  and  the  apparent  contour  of  the  given  object  its  base,  then 
the  section  of  this  cone,  made  by  the  perspective  plane  if  placed 
beyond  the  object,  would  be  larger  than  that  base.  That  is,  the 
perspective  of  the  object  would  be  larger  than  its  apparent  contour. 
If,  then,  there  is  any  error  in  the  projections  of  the  object,  this  er- 
ror will  be  magnified  in  the  magnified  perspective  thus  produced. 

Illustration.  See  Fig.  66.  Let  a- 
a'b'  be  a  vertical  line,  at  the  distance 
ca  in  front  of  the  perspective  plane  : 
and  let  EE'  be  the  point  of  sight.  Then 
by  (58)  the  visual  ray  Ea-EV  pierces 
the  perspective  plane  at  A,  the  per- 
spective of  aa1,  and  the  ray  Ea-E'#' 
pierces  it  at  B,  the  perspective  of  ab'. 
Hence  AB  is  the  perspective  of  a-a'b\ 
and  is  evidently  larger  than  the  latter 
line. 

142.  Shadows  of  trees,  and  other  ver- 
FIG.   66.  tical    objects.    Knowing  that  these  are 

parallel  in  fact,  when   they  fall  on  the 

same  plane,  they  might,  by  overlooking  Ex.  7,  Hem.  a  be  made 
so  in  the  drawing.  They  should,  however,  all  converge  to  one 
point,  so  that  in  the  picture,  shadows  of  posts,  &c.,  at  the  right 
of  the  vanishing  point  of  rays  will  incline  to  the  left,  while  similar 
shadows  on  the  left  of  the  same  point,  will  incline  to  the  right, 
when  the  light  comes  from  behind  the  observer. 

143.  Time  of  a  given  aspect.  By  knowing  the  direction  of 
vision,  the  direction  of  the  shadows  of  vertical  lines  upon  the 
ground  will  indicate  the  part  of  the  day  at  which  a  view  was  taken. 
Thus,  if  the  direction  of  the  shadows  indicates  that  the  vanishing 
point  of  rays,  R,  is  to  the  right  and  below  the  centre  of  the  picture, 
while  the  observer  faces  the  west,  the  picture  will  represent  a 


PICTURES,    AND   AERIAL   PERSPECTIVE.  116 

morning  scene  ;  or  in  summer,  and  in  high  latitudes,  the  same  posi- 
tion of  R  would  indicate  an  early  morning  view  to  an  observer 
facing  the  south.     These  two  cases  could  be  distinguished  by  th 
lengths  of  the  shadows.     Also  if  R  were  above  and  to  the  right  of 
the  eye,  the  observer  looking  south,  an  afternoon  view  would  be 
indicated.     This  is  a  point  of  some  practical  importance  in  reveal 
ng  the  aspect  of  dwellings,  &c.,  on  proposed  sites,  at  given  time 
of  day. 

144.  Light  and  shade.  The  intensity  and  distribution  of  shade 
upon  a  body,  depends  on  so  many  circumstances,  and  is  subject  to 
so  many  modifications,  that  its  exact  representation  in  real  cases 
must  be  mostly  an  art  of  pure  imitation.  A  few  points  are  here 
mentioned,  as  guides  to  the  observations  of  the  student. 

1. — The  light  and  shade  of  a  body  depends  upon  its  form. 
Double  curved  surfaces  (87)  have  a  brilliant  point,  or  point  so 
situated  as  to  reflect  the  most  rays  to  the  eye.  Plane  and  single 
curved  surfaces,  have  a  similar  brilliant  line.  On  all  bodies,  the 
line,  at  all  points  of  which  rays  are  tangent  to  the  body,  is  the 
apparently  darkest  line.  Its  exact  construction,  in  any  given  case, 
is  a  problem  of  practical  geometry,  of  more  or  less  complexity. 

2. — Light  and  shade,  depends  upon  the  nature  of  a  surface,  as 
dull  or  polished.  In  the  former  case,  the  brilliant  point  is  less 
intense  and  is  more  expanded  into  an  area,  while  all  parts  of  the 
body  towards  the  eye  are  distinctly  visible.  In  the  latter  case,  the 
more  perfect  the  polish,  the  smaller  and  more  intense  the  brilliant 
point,  and  the  more  nearly  invisible  all  the  rest  of  the  body,  owing 
to  the  absence  of  reflectinos  from  it,  directed  towards  the  eye. 

3. — Light  and  shade  is  affected  by  distance.  If  a  surface  is  in 
the  light,  the  more  distant  it  is,  the  darker  it  appears,  owing  to  the 
extinguishment  of  the  reflected  rays  from  it  by  the  atmosphere, 
and  floating  particles  therein.  If  it  be  in  the  dark,  the  more 
distant  it  is,  the  lighter  it  appears,  since  we  attribute  to  it  the 
increased  light  entering  the  eye  from  the  greater  depth  of  illu 
mined  air  between  us  and  it. 

If,  then,  a  surface  be  seen  obliquely,  it  will  appear  gradually 
darker  as  it  recedes,  if  it  is  in  the  light ;  and  gradually  lighter  as  it 
recedes,  if  it  is  in  the  dark.  Shadows,  in  like  manner,  are  darkest 
where  nearest  to  the  objects  casting  them,  and  lightest  in  their 
remotest  portions. 

Hence,  at  very  great  distances,  the  contrast  between  light  and 
shade  diminishes,  as  seen  in  the  faint  shades  and  shadows  of  hills 
in  the  misty  distance. 


116  LINEAR   PERSPECTIVE. 

4.° — Light  and  shade  is  again  affected  by  the  nature  of  the  light, 
A  diffused  light,  as  on  a  cloudy  day,  partially  confounds  lights 
and  shades  in  a  monotonous  uniformity.  A  concentrated  light, 
as  on  a  clear  day,  affords  well  defined  and  vivid  contrasts  of  light 
and  shade. 

But  again;  an  intense  light,  as  that  of  the  sun,  produce*! 
reflections  so  strong  as  to  diminish  the  contrasts  of  light  anj 
shade,  while  the  black  shadows  occasioned  by  the  weaker  moonlight 
are  in  familiar  contrast  with  the  white  lights  afforded  by  it. 

145.  Edges.     In  shaded  drawings,  edges  are  never  to  be  distin- 
guished by  black  lines.     Being  really  rounded,  through  imperfec- 
tion of  human  instruments,  or  by  attrition  or  crumbling,  they  have 
their  own  brilliant  lines,  as  cylindrical  surfaces,  and  are  distinguished 
by  lighter  tints  for  a  minute  width.     The  brilliant  lines  of  edges  in 
the  light,  are  due  to  primary  light  falling  on  them ;  those  of  edges 
in  the  dark,  that  is  which  separate  two  dark  surfaces,  to  reflected 
light ;  and  both,  to  the  superior  polish  acquired  in  part  by  the  fric- 
tion of  passing  particles  to  which  edges  are  exposed. 

Those  edges,  however,  which  separate  light  from  shaded  plane 
surfaces,  are  minute  cylindrical  surfaces  so  placed  as  to  have  a  line 
of  shade  (144,  1°)  upon  them,  and  may  be  indicated  by  a  line  of 
slightly  darker  tint  than  that  of  the  illuminated  surface  which  they 
bound. 

Hence  parallel  surfaces,  near  together,  should  not  be  distin- 
guished by  material  differences  of  tint  on  their  illuminated  portions, 
but  by  the  treatment  of  their  edges,  and  by  the  shadows,  if  any, 
of  the  foremost  on  the  one  behind  it. 

146.  The  Color  of  objects  is  modified  chiefly  by  the  color  oj  the 
light  by  which  they  are  seen ;  and  by  distance,  and  the  condition 
of  the  atmosphere.     To  do  justice  to  the  former  topic,  would  lead 
further  into  optical  discussions  than  is  here  proposed.     In  respect 
to   the  latter,  we  observe,  that  distance  causes  all  colors  to  be 
confounded  more  and  more  in  the  blue  of  the  atmospheric  depths 
through  which  they  are  seen. 

Trusting  that  these  problems  and  notes  have  now  been  sufficient- 
ly extended,  to  guard  the  geometrical  draftsman  from  doing  offen- 
sive violence  to  artistic  truth,  and  the  artist  from  doing  equally 
offensive  violence  to  geometrical  truth,  we  here  terminate  both. 


THE  END. 


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31  1942P 

FEB 


LD  21-100m-8,'34 


